According to [this source][1], 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][2].

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.

**Added later**: see comments below. Apparently when _Handbook of Graph Theory_ quoted this result, they omitted an additional assumption required for this result, that the graph be 2-edge-connected. This is why one needs to check original sources.


  [1]: https://books.google.com/books?id=mKkIGIea_BkC&pg=PA268
  [2]: http://doi.org/10.1002/jgt.3190130507