That is right. But let's get some more facts stated, and then more clear conclusions can be drawn.
President Madison did not “start” the second bank. It was primarily six men who were working diligently on establishing the second Bank of the United States.
These six were financiers John Jacob Astor, David Parish, Stephen Girard, and Jacob Barker. Also there was Alexander Dallas, who would become Secretary of the Treasury in 1814, and Representative John C. Calhoun of South Carolina. These men helped author the bill for the bank that was passed by the Congress.
President Madison did not agree with the initiative and vetoed the bill in 1815.
This combination of businessmen and politicians would not take no for an answer. When Congress reconvened in December 1815, they used a Presidential initiative to bring the bank back up for a vote.
In this next term, President Madison urged Congress to address the question of a uniform currency. State banks had ceased redemption of their notes, mostly because of a lack of gold and silver. Madison felt that the time had come for Congress to introduce the country to a more uniform paper currency. The secretary of the Treasury agreed and in his annual report noted that “a national bank would be the best and perhaps the only adequate resource to relieve the country and the government from the present embarrassments.”
But when a vote on the bank arose at the end of January 1816, Madison vetoed the bank legislation again.
Taking up the question once again in February, the House of Representatives considered still another bill to create a national bank. Finally, after more debate, the measure passed the House and Senate, with Madison signing the bill this time.
By this time, Madison and the government were no longer faced with war financing, but still had to deal with the issue of establishing a stable currency.
Madison was not a proponent of repaying state debt, but was concerned with currency stabilization.
But getting back to TJ and Alex for a moment, the former clearly disliked the northern finance-types, hated NYC when he had to live there (and, for those interested, if you go to Google Maps, plug in "59 Maiden Lane, NYC" and then click and drag on the little yellow guy in the satellite view, then put him in front of that address, it will give you a street view. As you look at the big "59" on the building, pan left and you will see a guy sitting on a bench and a plaque on the building wall in front of him: that plaque commemorates Jefferson's apartment).
At any rate, TJ did not like these city-types and really hated the speculators (interesting how BHO attacks the 'oil speculators' today, don't you think?). How much of the First and/or Second Bank were bad examples of crony capitalism I cannot say (although I'm sure there was some of that), but I also believe that just like Madison (an anti-Federalist cum Democrat-Republican), Hamilton's primary motive was to stabilize the currency and form a foundation for a solid economic footing for the early Republic.