Pilot reported a NMAC in the traffic pattern with an overtaking aircraft.

Narrative

Both aircraft left from XXR ZZZ1 for HYI (San Marcos) at roughly the same time after having lunch together. It was agreed that I would be first to fly and land so that the other aircraft wouldn't block me from taxiing to the hangar.Climb and cruise went well. ADSB showed the other aircraft in about a 2 mile trail. Descent into HYI also seemed to go fairly well. I tried to maintain a constant cruise speed during descent. ADSB seemed to indicate distance was being maintained. We both ended up getting clearance from the Tower for a left downwind entry to Runway 08; which more or less matched our descent heading. Tower asked that we report mid field downwind. I (Autopilot) leveled at 1000 AGL as we approached to enter the pattern. I began slowing from 165 kts. cruise speed to pattern speed of roughly 100 kts. ADSB was intermittently reporting other aircraft down this low. I reported mid field downwind and I seem to recall the Tower clearing me to land Runway 8. The following aircraft reported the same shortly after. I glanced at the ADSB display to see how close they were; but it wasn't displaying the other aircraft at that time and I was having to slow to <96 kts. to deploy half flaps; so I didn't have time to look around. With the Tower there; I assumed they would do conflict resolution.As I got to the approach end of the runway; Tower told following Aircraft Y that he was overtaking me and asked if he had me in sight. He said no. Tower told me to extend and following Aircraft Y to turn base to land. At this point; I saw movement out of the corner of my left eye and saw the following Aircraft Y on my left wing performing a descending left turn to base. Tower asked me if I had him in sight and I reported he was off my left wing. Tower; then cleared me to land #2 following Aircraft Y. I don't know how close we came; but we were both probably close to the pattern altitude. It seemed like he passed over me when he turned base; but I only saw him visually after I looked left to see the movement.