DA42 Flight Instructor reported a blown tire resulting in exiting the runway and taxing to a ramp for assistance from maintenance.

Narrative

Flight instructor had a previous flight at XA:00 am in Aircraft X and noticed a 'wobble' when taxiing and landing that was distinct and unusual. Crew preflighted the aircraft well and noticed a small bald spot on the right main but still had tread and well within limits to fly. Flight instructor noticed during this flight that it was taxiing and landing with a slight 'wobble' and decided to full stop at ZZZ for further inspection. When we post-flighted; we noticed the same small bald spot; that had not progressed at all. Nothing on the post-flight or the next pre-flight would indicate anything other than normal operations. Flight instructor wanted to use caution; though; and decided to only do short-field landings at ZZZ just in case. Crew on the XE:00 am flight did one touch and go at ZZZ1 with no issues; only a slight 'wobble' on takeoff. We decided to come back to ZZZ and do full stop taxi backs. On our first full stop taxi back; the crew heard a 'pop'. We figured right away that a tire had popped; ATC confirmed that they 'saw smoke' from one our tires. Crew decided it was safe to taxi off of the runway. Instructor took controls for this. We were able to maintain control easily and ATC let us know that they didn't see any additional smoke or hazards. Crew decided to taxi to Ramp. We called dispatch and let them know we would need help from maintenance and asking where we should go from ramp. We got no response from dispatch for approximately 5 minutes. When we got onto ramp; Tower let us know the tire was off the rim. Crew decided to full stop there as to not add any additional damage to the rim. Dispatch still had not gotten back to us; so the instructor gave the student controls so they could call maintenance themselves. We were disabled on an active ramp for about 5 minutes before maintenance arrived and towed us to the maintenance hangar.