Tower Controller reported they allowed an aircraft conducting a go around to fly at an altitude below the Minimum Vectoring Altitude.

Narrative

I was working Local Control combined with Ground Control; Flight Data; and Clearance Delivery. There was no other traffic in ZZZ. Winds were around 5 knts. with some mild precipitation over the field. Approach ZZZ Sector coordinate an Opposite Direction arrival for Flight Number to RunwayXX. I approved this request and the Aircraft X attempted an instrument approach to RunwayXY. At 1/2 mile final the Pilot reported a go-around. I asked the Pilot if they would like to remain under my control and attempt a VFR approach to RunwayXX; the Pilot decided they wanted to do this but wanted vectors. I then coordinated with Approach for control of the Aircraft X to be transferred back to Approach. I instructed the Pilot to fly RH while climbing to 030. The Pilot was south of the Runway at 015 heading NE on a heading of 060 at this time. The Pilot asked to be able to maintain 015 while on the 060 heading in order to maintain VFR while they make their Approach back to Runway XX. I allowed the aircraft to maintain VFR at 1500 ft. and instructed them to contact Approach and inform them that they are at 1500 ft. in order to maintain VFR. The Minimum Vectoring Altitude in the area of flight was 2000 ft. The Pilot went out to Approach and was climbed to 2000 ft. The Pilot attempted to a visual approach to Runway XX at 020 but was unable to join the final for the Approach. Approach then coordinated another Opposite Direction Approach to Runway XY. The Pilot then landed on Runway XY.In order to prevent this from happening again I will not allow aircraft to inform another Controller of their reasoning for a certain action. This is not an acceptable form of coordination in which assumptions are made and lead to improper flight handling. I will also not allow an aircraft to maintain altitudes below any listed Minimum Vectoring Altitude while leaving the airfield. I should have called Approach back with the Pilots request upon which they would have told me; unable; and had the aircraft maintain an altitude at or above the Minimum Vectoring Altitude.