It depends on the composition of protease inhibitor which you are using. Sometime it is a mixture of leupeptin, pepstatin and PMSF and if it is not then you can add. What generally I do, I add these protease inhibitors including PMSF separately.
PMSF is very unstable in water, so its really better to add it immediately before or after lysis. Like AEBSF, it is a suicide inhibitor, so once it inactivates a protease, that protease is gone for good. If your samples are very sensitive to proteases, there may be some small value in adding PMSF to a cocktail already containing AEBSF, as it may have nonoverlapping functionality (this is pure speculation, as we don't fully understand all proteases in yeast!)
You do not likely need to add PMSF, unless you are particularly worried about serine protease action and want to spike the mixture with extra insurance.
Hey Justin. Just a small comment on you saying that PMSF (or AEBSF) is a suicide inhibitor. From my understanding of suicide inhibitors (substrate-like compounds that form an irreversibly and covalently bound intermediate and thus inhibit the enzyme) PMSF isn´t one. PMSF just binds to the catalytic serine and thereby inhibits the enzyme. Or do I get sth wrong?