Here's some food for thought:
1. Lilly is usually sending you on an impossible number of missions as well as requests to improve the base. This leads to failed missions due to time restraints or a crappy base.
2. Missions do not run out if you are on another mission. In fact, time seems to freeze.
Hmmm... so I can take advantage of the little posses missions create in order to do side quests for resources, outposts, killing hordes, clearing infestations, and general murder? Sweet!
What I am referring to is the use of free buddies during missions to go complete all the other crap you usually save for when the missions have calmed down. For example, I just took an enclave that wanted to come back to my base (that's three badass helpers at once) to clear two infestations, build an outpost, call for a runner, and head back to base with a bag full of materials.
If I had done the mission to pick up those new members and then tried to complete the rest of my tasks, I probably would have failed the diversion and zombie hunt missions that Lilly was clambering about on the radio.
Use free buddies from the various missions in order to complete your side stuff. Makes the game more enjoyable, safer, and you'll complete more missions. I never, ever pay the 100 influence to have a buddy watch my back. If anything, when crap hit the fan that person would just end up getting shredded while I ran away! So save your influence and use your mission posses to kick zombie butt.