SMA TRAINING FLT TAKES EVASIVE ACTION TO AVOID CONFLICT ON APCH AS TWR CLOSING.

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INCIDENT OCCURRED WHILE ON A CHKOUT FLT WITH A CFII IN R SEAT. CHKOUT WAS FOR FLYING CLUB. I WAS PERFORMING NDB PRACTICE APCH (NDB 18) FOLLOWED BY ILS (ILS 36) DOWN TO CIRCLING MINS TO LAND ON RWY 27; WHICH WAS IN USE. REFS WERE BY INSTS ONLY (UNDER HOOD). UPON REACHING CIRCLING MIN; INSTR TOLD ME TO STAY UNDER HOOD AND HE WOULD VECTOR ME TO A L BASE FOR 27. TWR INSTRUCTIONS WERE TO ENTER MIDFIELD L DOWNWIND. INSTRUCTOR VECTORED ME ONTO DOWNWIND AND THEN TO A L BASE; AT WHICH TIME HE TOLD ME TO TAKE THE HOOD OFF AND PREPARE FOR AN ACCURACY LNDG. UPON TURNING APPROX A 3 MI FINAL; THE TWR INSTRUCTED I WAS #2 TO LAND FOLLOWING AN SMA ON SHORT FINAL. TWR THEN ADVISED ALL TFC THAT THE TWR WAS CLOSED FOR THE DAY. I BROADCASTED ON CTAF THAT I WAS FINAL FOR 27 TOUCH AND GO. AT ABOUT A 1/4 MI FINAL; THE INSTRUCTOR CALLED OUT TFC TURNING FROM A R BASE AND FINAL BELOW ME AND ABOUT 1000 FT AHEAD. AT THE SAME TIME; THE CTLR AT THE NOW CLOSED TWR ADVISED ME THAT TFC HAD CUT IN FRONT AND BELOW ME. ACFT WAS A HIGH WING AND TURNED FINAL APPROX 100 FT FROM RWY THRESHOLD AND ABOUT 200-250 FT AGL. I ADDED FULL PWR AND BANKED TO THE R IMMEDIATELY. I BROADCASTED ON CTAF THAT I WAS DEPARTING THE TFC PATTERN TO THE S. I NEVER HEARD ANY RADIO XMISSION FROM ACFT THAT CUT US OFF. I BELIEVE THIS INCIDENT OCCURRED BECAUSE THE TWR LOST TRACK OF ACFT; BECAUSE TWR HAD JUST CLOSED AND BECAUSE INSTRUCTOR PLT LEFT ME UNDER THE HOOD TOO LONG. ALSO; OTHER ACFT DID NOT BROADCAST INTENTIONS ON CTAF AFTER TWR CLOSED.