Traffic departing from FAI was issued traffic information but not informed that two of the same type were in the vicinity; resulting in a 360 degree turn and a potential conflict.

Narrative

I departed 20R VFR for MTO. There was inbound 1900 traffic for 2L. I requested on course and was cleared to fly 278 degrees on course and told to maintain 2000' for inbound 1900 traffic. I requested to turn North then for terrain avoidance and heard an 1900 cleared to land 2L. While turning I spotted a 1900 and told Tower I had the 1900 in sight. On Company frequency; I said to the Era pilot; 'I think I'm at your 2 o'clock.' The pilot told me he was on short final for 2L. I realized this was not the company aircraft out my window and I told the controller on 118.3 I needed to do a 360 to stay out of their way. The other 1900 then said they had me in sight and just got switched over from Approach and a few seconds later while I was in my 360; Tower cleared me to resume on course to MTO. The rest of our flight proceeded uneventfully. Cause: 1) There were two 1900s inbound to Fairbanks; not just one which was landing on the opposite direction of the runway we'd just departed. 2) The second 1900 was landing parallel to the runway I'd departed; opposite direction of what was landing toward us. 3) My personal opinion is that the Tower controller was not as experienced in FAI as some of the other longer term FAI controllers and so did not realize it would've been useful to know there were two inbound 1900s for Fairbanks and to convey that information. 4) The late switch-over or check on of inbound aircraft to Tower frequency. Tower acknowledged my 360; the [second] 1900 came up on frequency and said they had me in sight and stated they'd just been turned over from Approach. While in my turn someone said we were clear and Tower cleared me to resume direct on course. Suggestions: getting all aircraft within 3 minutes of airport onto Tower or the same frequency.