Finite state automaton for the reverse of the language, and multiple starting states
FSAs come in several varieties. The most common ones are deterministic (DFA), nondeterministic (NFA), and nondeterministic with $\epsilon$-moves (also called NFAs; sometimes $\epsilon$-NFAs).

The definitions of all three are not completely standard – there are several variants in the literature (for example, some allow the transition function of a DFA to be partial). However, in most standard definitions, all FSAs have a single initial state; though it is perfectly possible to define NFAs and $\epsilon$-NFAs having several initial states.

The reason you have to add a new initial state when reversing an NFA is exactly the requirement that there be a unique initial state. If we remove this requirement, then there is no need to add a new initial state.