A CRJ200 flight crew; after abandoning an unstabilized approach; descended too low while maneuvering for a subsequent visual approach and reported an EGPWS TERRAIN alert .

Narrative

Inbound we received a late approach clearance for the visual approach. It was a straight in approach. We were a little high and a little fast for the approach. As we crossed the final fix; I realized we were too fast and would not be stabilized for the approach. We made an early decision to go around. We notified Tower of our intentions to go around. He gave us an altitude and cleared us to circle around for a visual approach to the opposite end. A few minutes later; the Tower advised Approach wouldn't approve it. He gave us a heading and we switched to approach. Approach Control gave us a few headings and then cleared us for a visual approach to the other runway. As we started the visual; we got a little too low and received a 'TERRAIN' alert. We initiated a climb. Just as we initiated the climb; Approach Control stated he received a low altitude alert and asked if [we] had the airport in sight. We did have the airport in sight. We were then able to continue the visual approach and we landed. We were rushed on setting up and getting ready for the approach. Everything happened really fast so there were a lot of distractions in getting set up for the next approach...which resulted in us descending too low. One idea to prevent this would be to have better procedures and training to handle missed approaches and set up for the next approach. It would have been better for us to enter a holding pattern to allow us time to set up and brief the new approach.