David Tenner would probably be the best person to ask, as that man is a wizard when it comes to electoral statistics and such; I've been in awe of him since the SHWI days.
That being said, Hamilton certainly did his best to sink Adams, as the two feuded over many issues, including Hamilton's desire to go to war with France among over things. Mainly Hamilton just felt Adams to be too independent and wanted more influence in the administration and hoped to swing the vote to someone more open to his 'advice.' One of the major factors that hurt Adams was one of Hamilton's long letters that fell into the hands of a Democratic-Republican operative and greatly embarrassed Adams after it was published.
Now, if we remove Hamilton from the election somehow (lets say he comes down ill and isn't able to politic for the duration of the election) and thereby remove that published letter from history as well as many of Hamilton's efforts to weaken Adams, that I would say the President has a chance. However, there are still some very big obstacles for Adams to overcome if he hopes to gain a second term. Primarily, many of his policies weren't all that popular. The Alien and Sedition Acts, for instance, provided a wonderful point of attack for Jefferson against the Adam's administration, as did Adam's support of the Jay Treaty. Furthermore, in OTL, Jefferson won 61.4 percent of the popular vote to Adam's 38.6 which is a pretty large margin that we have to overcome.
There is one very interesting possibility here, though: New York at the time chose its electors in the legislature, which had been a Federalist stronghold. The Democratic-Republicans pushed for a change in the rules that would allow for the popular vote of Electors by District, since they felt they had no chance to gain control of the Legislature. When the Federalists refused, the Republicans used this as a campaign issue and carried off an upset, gaining a 2 seat majority and were, therefore, able to throw New York's electoral vote to Jefferson and Aaron Burr.
So, we could either have the Federalists agree to elect electors by the popular vote which might win enough goodwill that Adams carries the state. Or, we might also have the Republicans fail to exploit the issue and, although making gains, the Federalists remain in control of the Legislature and give the vote to Adams. In either case, with that shift of 12 votes, Adams wins the election.
But, what about the popular vote? What would be the impact of Adams winning the Electoral College while Jefferson wins the popular vote? Would Jefferson carry through on his threat to march the Virginia state militia on Washington DC if he lost? The early Republic was a pretty chaotic time as Americans adapted the Constitution and tried to come to some consensus about how the government should operate. Any of these scenarios could lead to some very interesting ramifications in the long run.