Flight Instructor reported confusion regarding a new taxiway configuration at PRC airport after recent construction changed the designations for certain taxiways not yet reflected on the airport diagram; only by NOTAM.

Narrative

When this safety event took place; it was the first time taxing to Runway 12 since the construction on Taxiway C which took place a few months back.After thoroughly completing the preflight checklist; before start checklist; and before taxi checklist I - the instructor - briefed the taxi diagram at PRC and the potential taxi routes that might be given. I was using Foreflight to access the FAA airport diagram and the chart was current with validity from October 5; 2023 - November 2; 2023.I also briefed about current NOTAMS at the airport; including the of renaming Taxiway C3 to F. I called PRC Ground requesting a taxi with the current weather and was told to taxi to Runway 12 - via D D3 Cross 21L at D3 and F to Runway 12. At this time I mentally noted that the controller did not say Taxiway C in the taxiway clearance.Once I received a taxi clearance; I briefed the route I was given and included in the briefing that when I entered Runway 21L from D3 Taxiway C3 (now F) would be straight ahead. As I taxied up to the hold short line on Taxiway D3 I queried Ground Control to confirm we were cleared to cross Runway 21L. Ground control reaffirmed I was clear to cross and began entering the runway.As I continued onto the runway; I was then able to see over to the other side - because of the curvature of the pavement - and did not see Taxiway F. In fact; there was no pavement directly across from D3 even though the FAA Airport Diagram depicted Taxiway F (C3) to be there. I quickly began to panic as I thought I might have been lost however I followed my clearance correctly and confirmed all taxiways with the airport signs and markings. I immediately began scanning for the Taxiway F signage and saw the signage to my left a few hundred feet down the runway. I turned left on the runway and began taxiing down Runway 21L. I then turned onto Taxiway F and continued to Runway 12.Even though I complied with the taxi clearance; there was a clear safety issue. The airport diagram should be updated as soon as possible and until then there should be a NOTAM to denote the change in taxiway location and the ATIS frequency should also note the different locations of taxiways compared to the FAA Airport Diagram.