With events such as the Non-Importation Act, continuing impressment, and just the ever existing threat of war with Britain, wouldn't that be the logical step to take?
Jefferson was not as anti-navy as he is often portrayed. The main issue with the raising of a navy initially was the inability of the government to raise the capital under the Articles of Confederation. Once the Constitution was ratified, the federalists argued that the cost of establishing a navy would be offset by the increased reliability in trade it provided through the protection of their trade vessels from impressment and piracy, as well as the bargaining power it offered in situations like the quasi-war. The Democratic-Republicans and other anti-navy advocates thought it was easier and cheaper to just pay tribute. The issue with Jefferson, as always, is that his ideological beliefs and his actions are at odds. Jefferson split with most democratic republicans in practice on this issue by advocating force when dealing with threats to Americans. In a letter to Monroe in 1786 he advocates force over ransom or tribute when dealing with these issues. The Federalists under Adams established the Naval department and expanded the navy considerably, so by the time Jefferson assumed office the American Navy was already large even when compared to what the pro-navy group had advocated. Pushing for any further expansion of the Navy by the time Jefferson was President would have been overreaching and politically damaging, especially because Jefferson had argued against such a large expenditure of borrowed money when the program was proposed. As president, he pared down the Navy which had more than doubled in size during the previous administration, in order to bring it more into line with what most viewed as a strong but reasonable force, rather than a force which was larger and more costly than was reasonable given the means of the government at the time.
Ellis, Joseph J., American Sphinx: The Character of Thomas Jefferson. 1996.
Julia H. Macleod, "Jefferson and the Navy: A Defense" Huntington Library Quarterly Vol. 8, No. 2 (Feb., 1945), pp. 153-184