The first thing to recognize is that there's a difference in "knowing the pressure" and "modeling the pressure." If the model is really good, then "modeling" does mean "knowing," but that can be hard or impossible to do sometimes. Below the transition point, the pressure is related to the temperature through the ideal gas equation of state -- the classic $pV = nRT$ or $p = \rho R T$. When you have a supercritical fluid, however, this equation of state doesn't work so well and it is no longer a good model.
Unfortunately, supercritical fluids are hard to model. Very hard to model. The most successful equations of state for them are called cubic equations of state. You can see that they add complexity to be sure, and they begin to introduce additional constants and non-linear functions. These constants are usually calibrated to experimental measurements and critical properties, but these have to be measured away from the critical points. Many thermodynamic terms are undefined at the critical points, making modeling very complicated there. Mixtures of gases are also very complicated to model well, because the equations of state for the individual components have different constants and mixing rules to combine them aren't universal.
More to the point, though, you said that you need to know more than just the temperature to find the pressure. This isn't the most precise way to say it -- yes, you do need to know additional constants and additional functional forms to relate pressure and temperature. But, no matter how complicated the equation of state becomes, any two state relations can be connected by a third! In all cases, if you have $V,T$ or $\rho, T$, then you can find $p$. It might be linear, like $p = \rho R T$, or it could be cubic like the more complicated equations I linked to. Or it could be some much more complicated functional model that hasn't been developed yet. But any two state variables can find the third.