Who was the most productive US president after they left office?

Andrew Jackson continued to wield an enormous amount of influence over the Democrat party after he had left office in 1837. Texas annexation had failed in Congress and Sam Houston was reluctant to request annexation again. It was largely on the assurances of Andrew Jackson of Senate passage that Houston agreed to resubmit his request to have Texas enter the Union. Subsequently Texas annexation passed via highly dubious (and possibly illegal means) on a largely party line vote. Jackson was also enormously influential in determining the Democrat candidate for the election of 1844. Martin Van Buren was the presumed nominee, being the chief architect of the Democrat party and the former president. Van Buren and his likely opponent, Henry Clay acting jointly, took very cautious attitudes regarding Texas annexation( annexation being something Jackson had also opposed during his Presidency). Largely on the maneuvering of Tyler's secretary of state Abel Upsher, Texas annexation became an issue of national defense and a means of defending slavery. Jackson had grown increasingly defensive of slavery in his retirement years, and with Upsher's political coup embraced Texas annexation. In doing so he reluctantly refused to endorse Martin Van Buren for the nomination. It would be stretching facts to claim that Jackson was solely responsible for Van Buren's defeat (plenty of Van Burenites voted for changing the convention rules which denied Van Buren's nomination on the first ballot) but it certainly played a large role. Ultimately James Polk would win the Democrat nomination, branding himself as a young hickory ( Jackson was old hickory, and the old warrior loved the reference to himself).

I go into some detail regarding the specifics of the political aspects of Texas Annexation in this comment

Edit: No idea if this meets your definition of being productive politically