First Officer reported local ATC amended the departure altitude which resulted in a low altitude alert from Center.

Narrative

We were scheduled to take passengers out of EGE. The initial clearance we received on the ground was to depart Runway 25 and climb via the BEVVR 1 departure RUBAY transition; the top altitude of which is 15;000 ft. We briefed our plan A and B; and the Captain made special emphasis on cleaning up the aircraft at 7;000 ft. (slightly above our normal level off altitude of 400 feet). We took off Runway 25 and made the turn to COPER. During the climb I was slightly late (about 50 feet) on making the 7;000 foot call out; so the Captain called it out and we cleaned up the aircraft. Around 8;300 feet; Tower called and advised us to maintain 10;000 feet; contact Center. I don't remember if it was Tower or Center; after checking in with them that told us to expect higher shortly. I set the altitude selector to 10;000 ft.; called out '10;000 set and 9;000 climbing 10;000.' Looking back I don't remember if the Captain acknowledged my call outs as required by our SOPs. This should have been an indication to me to verify the altitude with him again and to check with Tower to confirm the instructions. I called Center and advised that we were '9;000 climbing 10;000.' We had the TAWS page up prior to take off and we could see the terrain visually. We were in the black the whole time and didn't received any sort of TAWS alert. We leveled off at 10;000 and maintained the speeds required by the departure while we waited for a higher altitude as we had been previously advised. Between BEVVR and APRES we received a low altitude alert from Center and were cleared to FL260. Center advised us that the MIA (Minimum IFR Altitude) in that area was 13;100 ft. and told us that there was a required climb gradient on the SID for a reason. We of course began our climb to FL260; advised Center that we had been assigned 10;000 ft. and asked them to mark the tapes and provide us with a phone number so we could call once we reached our destination. I still don't know if the Tower Controller had misspoke or if I had misheard the altitude instruction; but a lapse of communication took place somewhere. I should have questioned the unusual altitude assignment with the controller and verified that the Captain heard the same instructions as I had.