hopefully you get some more doffs as you level up. perhaps a new batch at each rank up (commander, captain etc) assuming thats not already the case.
also i have yet to get here but do you get more slots for your ship and ground. ie more than the 5 you get at lt.cmd. if you dont i think it would be nice if the number increases as you level up to give you more options.
In SFA I got a 1 day assignment to recruit some cadets, and a 3 day assignment to recruit some officers. Beaming up to Sol Space, I got a colonial assignment to transfer some settlers where the reward was 5 (iirc) DOffs and some Dilithium. It's important to note that SFA, ESD, and Sol Space each count as separate "zones" with their own sets of DOff assignments. Also, look for the higher quality assignments. They're color-coded just like items.
What it sounds like is that the game needs to give better guidance on how to obtain more Doffs, having to hunt around for people does not replicate a well established organization like Starfleet. The Doff main page should list available missions for recruiting personnel (where and how).