Shkadov thruster gravity tug
We'll pull the SMBH with an array of Shkadov thrusters.
Finding star mass & Shkadov acceleration
Borrowing from HDE 226868's answer to Which type of star would be best used for a Shkadov thruster to reach Andromeda as soon as possible?, which asks what type(s) of stars might be best suited for intergalactic travel. It is determined that stars above ~2 solar masses have a terminal velocity of 0.045 c. This is because gains in stellar luminosity and Shkadov thrust are bogged down by gains in stellar mass and limited by stellar lifetime. Making stars bigger has little effect on top speed, however, as we'll later see, it has a big effect on travel time.
In the same question, L.Dutch's answer extracts acceleration proportionality coefficients for various star types, based on the idea that acceleration is proportional to luminosity. A one-solar-mass Shkadov thruster can reach ~20 m/s delta-v inside 1 Myr, for an acceleration of $6.35 \cdot 10^{-13}$ $\text{m/s}$. We can multiply L.Dutch's coefficients by this baseline acceleration to infer accelerations for higher-mass stars. A 2-3 solar mass star might seem a reasonable place to start; however, cranking the numbers, the time it takes one to reach its top speed of 0.045 c is greater than the age of the universe by about 2x, and stellar lifetime of such stars is only around ~400 Myr.
From this information alone, it becomes impossible to make a Shkadov thruster perform on reasonable timescales without the assumption that stars can have their lifetimes artificially extended by technological means, i.e., removing the "rot" of heavy metals from their cores. (Suppose a powerful beam (perhaps redirected from the Shkadov thruster) can drill a hole down to those inner regions, clearing a channel for the heavy element plasma to "leak" out, in part due to great pressure.) I will proceed under that assumption.
(Note: increasing the star's lifetime likely changes our initial assumptions about the top speed. For reasons discussed later though, it might not matter much.)
Now that we have the ability to prolong star lifetimes, it's perhaps reasonable to jump to the most massive star on L.Dutch's table. A 60 solar mass star can reach the top speed of 0.045 c in ~21 Myr, with an acceleration of $2.03 \cdot 10^{-8}$ $\text{m/s}$. (Note: this will NOT be the SMBH's rate of acceleration, too.) The star will die in a spectacular supernova in only 3 million years, but by removing the heavy elements and keeping the star properly fueled, it will burn for much longer.
Initial look at Shkadov tug performance
The next step is figuring out how to arrange the Shkadov thruster to tug the SMBH. I do this using Newton's law of universal gravitation:
$$F=G\frac{Mm}{r^{2}};$$
Where, $M$ is the SMBH mass and $m$ is the star's mass (the rest should be familiar). We want to find the distance, $r$, to position the 60 solar mass star in such a way that the gravitational acceleration of the star is balanced by its own Shkadov acceleration, such that it neither falls into nor pulls away from the SMBH. To do this, we simply set $\frac{F}{m}$ (acceleration of the star towards SMBH) to the Shkadov acceleration found earlier, $a=2.03 \cdot 10^{-8}$ $\text{m/s}$, and plug, say, 1 million solar masses for $M$.
Rearranging the equation, I get a general formula,
$$r=\sqrt{G\frac{M}{a}},$$
And a distance of around 540,000 astronomical units (AU), or 8.5 light-years. The gravitational pull of the star on the SMBH is 4 orders of magnitude weaker, an acceleration of $1.22 \cdot 10^{-12}$ $\text{m/s}$. This is the acceleration of the SMBH towards the destination, and at that rate it'll take infeasibly long, over 350 billion years, to reach 0.045 c.
What we need is more mass pulling on the SMBH. To get the full Shkadov acceleration, we'd need another SMBH's worth of these thrusters stationed at the 8.5 light-year mark, approximately 17,000 stars of 60 solar masses. If you could command that many stars, you might as well forego the whole "tote the SMBH across a sliver of the universe" and just travel to the other galaxy and make another SMBH.
Building a proper gravity tug
Let's assume you're able to gather 1/50th of the previously mentioned number of stars. 340 in total. Such a mass would impart $4.14 \cdot 10^{-10}$ $\text{m/s}$ acceleration on the SMBH and reduce the 0.045 c delta-v expenditure time to ~1 billion years. If half the delta-v is spent on acceleration and half on deceleration, the SMBH could travel 11.6 million light-years (Mly). If the SMBH never performed deceleration, it could travel over 23 Mly.
1 billion years to reposition over 1,021,600 solar masses nearly 12 Mly is really not that bad. Keep in mind this is using several hundred 60 solar mass stars, each accelerating harder than any other Shkadov thruster in the entire universe.
But the question still remains, How would you position all these enormous stars so closely together?
You could be erratic about it and build something like a mini globular cluster, hope none of the stars crash into each other in 1 billion years. And even if some do, it's not a big deal. Maybe you can handwave something like the Shkadov thrusters redirecting some of their thrust to prevent collisions or ejections.
Or, you could assemble them into a semi-stable orbital configuration, such as Jenkin's toroidal swarm configuration.
The configuration consists of objects on slightly eccentric, slightly tilted, and slightly offset trajectories. This arrangement has the benefit of keeping "neighbors next to neighbors", so no high-velocity near-misses, and is relatively stable.
Due to the mass of the torus itself, the elliptical orbits will precess over time. ... That will happen after thousands of orbits, give or take a few orders of magnitude depending on the total mass of the torus relative to the inner sun. It can also be caused by an oblate sun (which happens if the sun is spinning rapidly). However some well-placed masses inside or outside of the torus can prevent the near point from precessing out of the central plane, eliminating this problem. There is also precession of the whole torus around the central axis, but the torus is already symmetric about the central axis so this is not a problem.
Those "well-placed masses" are not depicted in the above diagram, but are in this one:
A ring of objects orbiting within the torus helps to keep the configuration stable and adds to the coolness of it all.
The swarm configuration could be light-years across, taking many thousands of years for a single star to complete a single orbit, and the central mass perhaps being a more tightly bound cluster of thrusters. Intersections of a star's orbital track with the drive plume of another Shkadov thruster are unlikely due to the vast distances, and are probably harmless anyway. Over 1 billion years, the thrusters could complete a million orbits and so orbital corrections, possibly by redirecting some thrust, would be required for each thruster.
All in all, roughly to-scale the whole thing would perhaps look like:
Afterword
I'd like to point out the ludicrous amount of energy we're looking at here.
The SMBH of 1 million solar masses has a mass-energy content of 1.8E53 Joules. At top speed, the SMBH has a kinetic energy of 1.9E50 Joules, only 3 orders of magnitude less. It requires twice that to decelerate again. That's nearly 4,000 solar masses worth of antimatter annihilation energy packed into sheer speed. Four. Thousand. Solar masses.
Put a different way, that's the complete energy output of over 30 million Sun-like stars for over 1 billion years. To get up to 10% c like you initially wanted would require nearly 200 million suns' worth of combined energy for 1 billion years. That's basically a galaxy's worth. It's also nearly 11,000 solar masses worth of antimatter annihilation energy...
