I had trouble finding suitable assignments to keep maxed at 20 as well. At times it was hard to keep 20 assignments rolling at once all the time. I found that every where I went had two lvl gated missions most of the time. Generally they involve purple data samples as a reward. I also had nothing doable shipboard wise for a longtime Saturday night into Sunday. I was looking at two lvl gated missions and two reassign missions and that was it for a long time .
I was scrounging sectors only to find missions to be level gated or needed a doff that I didn't have (not on duty or on another mission, outright did not have) or a tribble that I didn't have. I found that there are lots of doff types that I am still missing even though my roster is ~60 of 100... I have six Tama's, not even joking... but I don't have a biologist.
I have been taking advantage of the 6 Tama's as I have started using the commodity cheat sheet posted here on the forums to visit all correct vendors to get stuff on the cheap. I then game the system with the "Trade commodities For EC" missions because the EC you get is much higher than the prices of the commodities sold at right vendor.
Found a cool mission at the captains table, you should put some special ones there...