Given the lack of opposing naval forces after the 1996 GIUK sea battles, I'm less than convinced much effort would be made to immediately attempt rebuilding the US navy. Production priorities re steel, electronics, manpower, etc are more likely to be redirected towards AFV production and raising ground force units (with perhaps a slightly lesser emphasis on air power).
As Olefin indicates, reserve ships should probably suffice for the foreseeable future from a 1996 viewpoint at least with more modern designs and production halted until the expected end of the war in a year or two (at the time NATO was doing very well and it seemed like only a matter of time before the PACT was defeated - then they used nukes...).
What resources were allocated to naval forces would probably be directed towards repairs and upgrades at that time. Come late 1997 however those decisions may well have been regretted, but regardless, it would seem quite improbable that more than a handful of new ships would have been built, even if the decision to do so had been made early enough.
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