As I remember, one role of epsilon-transitions in the theorem about the equivalence of regular expressions and finite automata is to implement alternation |. Suppose you have two properties P(s) and Q(s) and two automata AP and AQ that compute those properties. If you need to compute the disjunction of P(s) and Q(s), you can make an epsilon-transition to both automata.

If you have concrete examples that need analyzing, feel free to post.