How to traverse on only a cycle in a graph?

I'm attempting ch23 in CLRS on MSTs, here's a question:

Given a graph G and a minimum spanning tree T , suppose that we decrease the weight of one of the edges not in T . Give an algorithm for finding the minimum spanning tree in the modified graph.

A solution I found was to add this new changed edge in T, then exactly one simple cycle is created in T, traverse this cycle and delete the max-weight edge in this cycle, voila, the new updated MST is found!

My question is, how do I only traverse nodes on this simple-cycle? Since DFS/BFS traversals might go out of the cycle if I, say, start the traversal in T from one endpoint of this newly added edge in T.

One solution I could think of was to find the biconnected components in T after adding the new edge. Only one BCC will be found, which is this newly formed simple-cycle, then I can put in a special condition in my DFS code saying to only traverse edges/nodes in this BCC, and once a back-edge is found, stop the traversal.

Edit: graph G is connected and undirected btw