First, try to take over the GOP. If that fails, a new party is the only option save not voting at all.
While N.Y. does have the Conservative Party and they do put up some candidates for N.Y. state offices only one was elected to Us. House or Senate and only for ONE Six year term
James L. Buckley 1971-1977 Conservative
Now looking at the N.Y. Senatorial history I see MANY third parties.
For states with OPEN primaries the option of multiple parties may very well have the same results as N.Y. State