A light aircraft conducting flight training experienced an NMAC with a light twin in the pattern at DAB.

Narrative

We were conducting flight training at DAB in closed traffic on Runway 4R. There were approximately 6 aircraft in the pattern at the time with more trying to enter the pattern at higher altitude. I was told to go around on landing; which I did; climbing straight ahead to Traffic Pattern Altitude. The same time at the go around; a light twin called mid-field downwind. On the climbout; an aircraft entered the traffic pattern going the wrong way. When this occurred; the light twin made a left 360 degree turn to avoid traffic. At that time; I cleared my wing and made a turn to crosswind. That's when the midair almost occurred. The light twin re-entered downwind right where our crosswind intercepted it. Evasive action was taken and after this the Tower told me to follow the light twin traffic. I think better situational awareness on the Controller's part and more crew reports could help this from recurring. Maybe a rule to enforce the number of aircraft allowed at a time in a pattern would limit the potential for accidents.