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