A. Do as many pullups as you can, then switch to pull downs.
B. Instead of pulldowns you can get a jump stretch band from www.elitefts.com and use that for assistance in the bottom possition of the pullup (get a light or average band).
C. When I was in the Marines, we’d “train” pullups at almost every PT session by doing sets of 10 or max sets. But in my oppinion, I was able to easily get 20+ on test days by my workouts in the GYM, not necessarily on the bar.
On my “back day” I’d pick a number like 50, then do weighted pullups in sets of 5 (or triples/doubles etc) until I reached 50 pullups. I’d usually end with just body weight, but the goal was to get at least one more weighted pullup out of the 50 total each week. Then I’d do deads, rows, snatches, more rows, maybe some pulldowns, whatever. I think that helped more than anything.
Hope this helps.