In Jim’s forever book he has the 5/3/1 Original program with PR sets and he list the assistance work as 100 reps for Push/Pull/Single-Leg & Core. How would you break up that assistance work so it does not affect the next training day?
I do something like:
One a upper push day, choose assistance like for “push” that are relatively easy: pull downs, push ups, etc… Choose something like DB goblet squats (5x10) and ab roll outs (5x10) for lower/core, and then do pull ups and rows to get to you number on pull.
This should NOT affect your barbell training, and if it does you will need to get to the point where it does not. You will increase your work capacity in running the program.
That said, if you find that doing, let’s say, dips affects your bench press the next day, then start with lower numbers and gradually build up to the number prescribed. When I first did 531, I couldn’t handle the DL volume. I did lower than the prescribed reps and gradually upped the number so I could handle BBB sets with it.