Or how can you modify (temperature, pressure etc.) of hydrogen or a different gas to get more "lift" in airships, if that's even possible?
Hydrogen is the lightest gas there is. In its common diatomic form (H2), it consists of two protons and two electron, making it extremely light, even compared to the next lightest gas, helium (although the difference in lift between them is minimal). There's nowhere better you can go, unless you choose to simply use Hydrogen-1. Plus, if you heat it up a bit, then - just as with hot air - it will provide more lift, as per the ideal gas law.
Okay, fine. You could go with a pure or partial vacuum (see also here), but this is typically considered difficult because of the extreme pressure difference between the inside and the outside. However, it has been considered, and has a chance of providing at least some lift.
But stay with hydrogen, if you're willing to put up with the risk of ignition.
Did I mention that hydrogen is really explosive?
Image of the Hindenburg disaster in the public domain.
The lift is not actually provided by the lifting gas, it is provided by the air it displaces, so lift can't exceed air density. The lifting gas simply provides the pressure to sustain an air displacing structure with large volume with minimal mass. The differences in lift between lifting gasses are actually simply the differences in the weight they add. Helium and hydrogen are already fairly light, so it is difficult to replace them something that weighs significantly less.
you could heat the gas within the bag, but this has a few problems. first, it makes it harder to contain the gas. this is especially true for hydrogen, which already is almost impossible to totally contain without leaks; which is a problem, because you would presumably want to use the lightest gas that you could.
the obvious solution here: hot vacuum balloons
If you're willing to get a little weird with your physics, nobody I know of has ever proved that negative mass isn't possible. I.e. matter that is gravitationally repelled by positive-mass matter and attracted to other negative-mass matter like a magnet.
We've never found any (and how would we? Any that was still left in the Milky Way when it formed would be heading out faster than we could catch it!) but it's ability to 'fall up' would give it greater lifting power than any positive-mass gas.
I thought of plasma. Not a gas, strictly speaking and, when researched on it, it happens to be discarded in the WP article about this topic. You would need an elastic material that resists the huge inside charge (avoiding ions to get electrons or electrons being got) just to get a slightly higher lift.
Another medium that in theory could be used is a plasma: ions repelling each other could give a pressure intermediate between vacuum and hydrogen and hence that counteracts the atmospheric pressure. The energy and the containment requirements are extremely impractical, so that it may only be interesting for science fiction.
Vacuum is not a gas either, but all you need is a extremely light, airtight and rigid material.
Anyway, both solutions would be even more (much much more) dangerous than hydrogen. Any tiny crack in the cover would result in a huge instantaneous and devastating ex/implosion.
And, of course, you also have the theoretical negative mass that nobody can be sure how would behave, what framework and energy order of magnitude you would need to handle it, how to get it or even if exists.
TL;DR: your biggest problem isn't the filling, but the container. Any society with the knowledge to make such a floating device would have for sure the technology to fly in a faster, cheaper and more secure way.
The only thing lighter than hydrogen is... a vacuum airship.
That's right, you can evacuate all of the air inside your aircraft to make it even more buoyant, but the problem is that the outside atmosphere will be pushing in with 14.7 psi so you will need a very strong air-frame to resist that force. Very strong air-frames are also very heavy...
But! The square cube law actually can come to the rescue here. As you scale the aircraft up, the volume of vacuum increases to the third power, while the surface area you need to build only goes up by the second power. This means there's a certain point where you can build a vacuumed out airship that will actually fly, despite its tremendous weight.