When you say SolarWinds lives on your C:\, is this also where the database lives?
If so, you could move the database to your E:, and leave the program files where they are. You'll need to stop all SolarWinds services, make the necessary changes in SQL to move the database location, then run the SolarWInds configuration manager and re-point SolarWinds to the new database location.
If not, and it's the program files that you need to move, then your only course of action is to remove it, and re-install on the E drive. If you do this, make sure you have a backup/documentation for any changes you've made to the website itself (custom logos etc), as when you re-install it'll recreate the Orion website.