Pilot of small aircraft reported an NMAC in the traffic pattern when they exceeded traffic pattern altitude by 500 feet.

Narrative

I was assigned takeoff clearance from the tower. The runway I was using was 18. Part of the clearance was to make left traffic and depart on the left downwind. As I was climbing out; I heard another aircraft that was a student pilot solo call the tower for takeoff clearance. Tower cleared him for takeoff; which I assume was to make left traffic as I believe the student stayed in the traffic pattern. I make my crosswind turn roughly 1900 ft MSL. I then turn downwind at 2100 ft MSL. Mistakenly I let the aircraft climb higher than 2100 ft and up to roughly 2600 ft MSL. The student pilot solo was in a light aircraft and what I assume made his crosswind turn before me. I was carrying passengers for this flight; so our climb performance was slower than normal. Tower calls and tells the student pilot to be aware of a Aircraft X in the downwind. This was incorrect as I corrected the controller twice on the ground that we are a Aircraft X. I do not remember the student pilot's response to the call. Then the Tower tells me about the traffic at which time I could not see because my aircraft was above him. Being a low wing; I had little to no ground reference. My passenger was able to call him out as he had a better viewpoint from looking from the back of the aircraft. I see that the aircraft moves to cross underneath me and heads away from me at 030 degrees. I then deviate to the left and advise the controller that I now have the traffic in sight. Earlier I stated that I was looking for traffic. I resumed my climb out to get a greater vertical separation from his aircraft and deviate to the northwest when my original departure was the northeast. I assume our vertical separation was roughly 300 to 500 ft. Our horizontal distance was roughly 200 to 300 ft in front of us. I continued to maintain separation from their aircraft and continued the flight like normal. I could have developed a better situational awareness by focusing more on the radio calls the Tower and student pilot were making. I also should have held my altitude so I would have spotted the aircraft sooner allowing me to have a higher reaction time to make a proper decision to avoid the aircraft. I had a healthy number of hours of rest roughly 7 hours. I was hydrated and healthy.