The two main ways I envisage a larger army and navy would be:
- Some Caribbean islands rebel alongside the 13 Colonies and so a navy is required to defend the island states and territories of the USA - though this might change the USA politically so-much that it's unrecognisable, though I doubt that.
- Alexander Hamilton becomes President after Washington.
A war with France or Spain isn't too difficult:
- Have the Quasi War go worse.
- IOTL the slave-owners in Saint Domingue tried to gain support from the Americans in the event of the abolition of slavery, if Hamilton, of a more anti-French Jefferson were President then they could guarantee that and follow through if the French still abolished it.
- In 1804 Spain denied American merchants the right to sell and ship in New Orleans, which caused a diplomatic break until the Spanish restored the rights a year or two later. You could theoretically have a Spanish-American War over New Orleans, which could expand into a general war over Louisiana, Florida and perhaps even Cuba.
I think overall the best way for these scenario's to happen is to have Hamilton become President, as he was in favour of a larger army and navy, and was anti-France and relatively anti-Spanish.
The only way I could imagine the third one happening would be, say, Hamilton creating a larger army and navy, then Jefferson taking over as President and having an alternate Anglo-American War, but with the Americans in a much better state to fight they might be able to win a victory - though I'm unsure if they could gain much territorialy.