J3 Cub pilot reported their engine lost power on initial climb and they returned to the departure airport.

Narrative

I arrived at ZZZ this morning with the intent of flying my J-3 Cub to work at ZZZ1. I filed my flight plan with ZZZ Center; I performed a thorough pre-flight inspection; including sumping the fuel tank and gascolator drain. I added a quart of oil. The engine started (hand-propping) without any issue. I taxied over to the fuel pump and topped off the tank on AvGas. I added 5.7 gallons. I then taxied to Runway XX. I performed an extended run-up based on how long the aircraft sat since its last flight. There were no pre-flight abnormalities. Both magnetos checked normal as well as the carb-heat check and idle check. I verified my pre-takeoff checklist prior to taking the runway.The runway was 2;505 feet long asphalt. I took off Runway XX.I performed an normal takeoff. The airplane rotated prior to indicating an airspeed (40 MPH) and appeared to be running normally considering its 65 HP engine on a hot summer morning. Upon reaching approximately 100 ft. AGL; the engine coughed and stopped producing power. I estimated there was approximately 500 feet remaining below me and the runway ends abruptly at a fence; followed by a road; shopping center; another road; and neighborhood. There was a field to my left; so I immediately pushed the nose forward for airspeed and turned left for the field. Suddenly; the engine began producing power. I took another left turn; and attempted to climb and fly a very close-in left downwind in an effort to maximize available landing distance on the pavement. The engine again stopped producing power. I pushed the nose forward and aligned myself with Runway XY about midway down the runway. I landed somewhat fast and by the time I touched down; the engine was producing power again. I taxied off the runway without incident.After shutdown; I sumped the fuel again and didn't see any contamination. I started the airplane again and taxied to the hangar without incident.I am unsure of the cause of the engine failure. It appears that it only happened in a climbing attitude. This same thing happened to me once before about two years ago; in this airplane; while in a climbing attitude.