So I'm trying to update the User Information List in SharePoint (hidden list). My User Profile Service and Mysites are all synced correctly with the most up-to-date information. Is there a particular Timer Job that will update the User Information List though?
There are two timer jobs that push data from the UPSA to the UIL.
User Profile Service Application - User Profile to SharePoint Full Synchronization and
User Profile Service Application - User Profile to SharePoint Quick Synchronization are responsible for this. That said, sometimes the sync stops working correctly. In that case, you may want to try running
stsadm -o sync -deleteolddatabases 0 which will reset the sync status and when one of those timer jobs runs, will update the UIL correctly. You can also get the PowerShell version from Replacing stsadm -o sync with PowerShell.