return all strings contains given string in suffix tree
Yes, you are right. You need to consume the symbols of $T$ and all occurrences of it shall be in the sub-tree below the position you stopped.

Since the edges are stored in lexicographical order, you can perform a Depth-first to print all the substrings of $S$ starting with $T$ in lexicographical order.

Using this example from Wikipedia

and assuming $T = A$, we would arrive at an internal node. Doing a DFS, we would have: $A\$$, $ANA\$$, $ANANA\$$.