According to this source, a graph with a unique 2-factor must have a degree-two vertex. The result is credited to
Jackson, Bill; Whitty, R. W. (1989), "A note concerning graphs with unique $f$-factors", J. Graph Theory 13 (5): 577–580, doi:10.1002/jgt.3190130507.
However I don't have subscription access to that paper to verify that it really says that. So anyway, yes, if this is all accurate, then graphs with $\delta\ge 3$ have a second 2-factor when they have one.