So, I was wondering if for small scale weaponry (150mm guns and smaller) using a coilgun to assist in the acceleration of a traditional chemically accelerated round, either while in the barrel burning the propellant, or after the round has exited the length of barrel required to burn the propellant.
It could be used to supplement the speed of a chemically propelled weapon but unless you like the flavor of a hybrid launcher it's probably impractical compared to a strict magnetic or strict chemical system.
The kinetic energy of a projectile is proportional to the square of the velocity. If you have maxed out the energy that you can add to a projectile via a chemical propellant to get a 50% increase in the velocity you'd need to put in more than double the energy. Unless you absolutely need the extra velocity, you're adding an entire extra system that needs to be powered, maintained, and carried for a nebulous benefit.
If modern firearms are any indication getting a minor increase in velocity isn't going to be worth it.
In principle, you could, however there are going to be a number of practical considerations. For the lengths of the weapon you stated, (~150mm- I'm going to assume this is the total length as opposed to barrel length) you wont have a significant amount of space to fit your coils, and on top of that you are going to need to keep your power source. Considering that the amount of energy you can impart to the projectile through EM fields is related to the length of the barrel, you would likely not gain any substantial increase in muzzle velocity.
If it did work at all, you would get a small increase in velocity at the expense of a substantially heavier and larger weapon, when you in all likelihood could have achieved the same effect by user 'hotter' ammunition.
Frictionless barrel + afterburner oomph
Though the cost of power switching and other factors can limit projectile energy, a notable benefit of some coilgun designs over simpler railguns is avoiding an intrinsic velocity limit from hypervelocity physical contact and erosion. By having the projectile pulled towards or levitated within the center of the coils as it is accelerated, no physical friction with the walls of the bore occurs
Physical contact and friction is an issue with chemical firearms too though arguably less than with a railgun.
I could imagine a hybrid setup with a snubnose pistol type chemical firearm. This accelerates the projectile into the coilgun which accelerates it further. Acceleration conferred by any firearm is proportional to the barrel length. The cool thing about a coil gun is that regardless of how fast the bullet is moving when it enters, the coilgun can accelerate it faster.
A superconducting coilgun called a quench gun could be created by successively quenching a line of adjacent coaxial superconducting coils forming a gun barrel, generating a wave of magnetic field gradient traveling at any desired speed.
Good old 1/2mv^2. The wallop packed will increase as the square of the added velocity.
Of note - a coilgun that worked like an induction brake need only induce a current within the projectile to make it into an electromagnet. Your bullet does not need to be a magnet itself.
The big question - it is possible to impart spin on a projectile as it travels in a vacuum thru a coil?