I recently took on a role like this, where they had on-site folks doing Tier 1 type work but everything more significant was outsourced. I started out with some pretty solid goals on what I’d like to accomplish, but not long after getting deeper into the environment did I find out that I discovered there were some pretty significant and severe issues with the infrastructure that the previous service provider had simply been band-aiding over and over again. This led to things being in a really terrible place (like password changes and mailbox creations wouldn’t stick due to severe AD corruption).
I think probably the best thing you can do to start with is listen to your users and try and establish patterns. Try and understand the biggest pain points and once you’ve been in the mud for a month or so figure out what you can attack with the budget you have and what folks are going to have to live with for a while longer. Try and dig into any recurring issue and find the root cause. Don’t fill your time doing menial tasks just to look busy. Build the processes that they won’t have in place for requesting help, equipment, etc. and make sure you’re on course for best-practices like standardization and license compliance.