Hidden symmetries in second-class constrained systems: Are new fields necessary?

For many systems with second-class constraints, the question posed in the title is answered in the negative. We prove this for a range of systems with two second-class constraints. After looking at two examples, we consider a fairly general proof. It is shown that, to unravel gauge invariances in second-class constrained systems, it is sufficient to work in the original phase space itself. Extension of the phase space by introducing new variables or fields is not required.