The USN never possessed 671 warships. It possessed, ships of all types totalling 671. The total includes receiving ships, hulks, colliers and part built ships on the slipways. Most of the navy were converted merchant ships used to police the blockade of the CSA and a great many of them were small sailing schooners of less than 100 tonnes displacement, many more were converted tug boats. In total the USN had around 100 actual steam warships of which less than thirty were ocean going. Most of the civil war gunboats were made with green wood and were already deteriorating at the end of the war. Some of the larger steam sloops of war were in a similar condition. The double-ended gun boats were so fragile that a few rounds would destroy them. All of these warships having wooden hulls are effectively obsolete by 1865. The Union has one effective ironclad warship the USS New Ironsides, she is almost the equivalent of the British and French floating batteries built for the Crimean war. The monitors are a technological dead end, none of them can survive a sea battle not even the big ones, and they are frankly more dangerous to the crew than to a real warship.
It is actually hard to tell if the USN was the second largest in the world in 1865. It might have been, it might not. It is unquestionably smaller than the RN, that can be checked against the Navy lists but it is harder to tell if it was bigger than the French or Russian Navies because modern treatments of these navies do not tend to mention yard craft, receiving ships tugs, small schooners being used as tenders and so on as they are too small and insignificant to bother about being of no military value. Treatments of the USN tend to mention everything that floats no matter how small.
In terms of capability the USN is probably somewhere around 8th in the world in 1865. Clearly it is far behind Britain, France, Russia, Austria-Hungary, Spain, the Netherlands and Italy which has just come into being. It is also probably behind the Ottoman Empire and perhaps Prussia and Denmark. It does not slip behind the South Americans until the 1870s and even then not for very long.
So to answer the question the USN never had that much in the way of naval might after the ACW even though they had a large navy. They can keep the large navy if they want to by cycling out and replacing the ships that are falling to bits with newer merchant ships and few new constructions but why would they there is no one to blockade.
They can also if they wish build a completely new navy of high capability but they will be playing catch-up with the Europeans for at least 20 years and the British even longer. It will cost them so much money it will retard the development of the country.
They could do a little better than OTL. Get rid of everything but the serviceable real steam warships immediately including all the monitors and buy a new iron hulled ship every year and a real ironclad (not a monitor or battery) every three or four scrapping a steam warship of the old navy every time a new one is launched. They can afford that. It is probably about the most affordable way to a better navy but it is not going to be world class for years.