In my own modest opinion, I would say - if anything - it reflected rather than contributed to antebellum instability. Also, a healthy dose of blind luck. Statistically, with such a small n sample (15), one or two outliers radically impact outcomes. Same principle here.

1. The first seven presidents survived their term of office. Among the second set, TWO died in office. Consequently, two man became President for whom nobody particularly cared and who stood no chance at reelection. That's basically, like, 50% of your sample that is basically historical accident.

2. The first period featured the collapse of the two party system and over a decade of one party rule.

Mostly, it's probably #1. But, from an instability point of view, the second period featured the implosion of that single party rule from your first period. That cries electoral instability, and the elections (maybe?) reflect that. Additionally, the first period (1791-1837) is basically the last days of a ye-olde early modern nation state. The second period features the rise of industrialism and the explosion of the market economy.

1. Washington (2 Terms. Also, he is Geo Washington and he gets to be in charge for as long as he wants to basically no matter what. You might as well start your count with Adams.)
2. Adams (1 Term, super close election)
3. Jefferson (2 Terms.)
4. Madison (2 Terms, basically running against a rump Federalist party with no home)
5. Monroe (2 Terms. Speaking of historical outliers: UNOPPOSED.) ---begin market economy, panic bubbles, collapse of one party rule, aggressive expansionism, and beginning of snowballing abolition movement---
6. John Quincy Adams (Doomed to one term basically no matter what)
7. Andrew Jackson (2 Terms)
8. Martin van Buren (1 Term, loses because of economic panic)
9. William Henry Harrison (Dies almost immediately)
10. John Tyler (Doomed to one term basically no matter what)
11. James K Polk (1 Term by choice)
12. Zachary Taylor (Dies)
13. Millard Fillmore (You know, Nathaniel Hawthorne's roommate from college.

I mean, I could keep going but it's pretty clear that second set has a lot of historical accident in it and I have things to do. In the case of the one termers it is generally that instability drove them from office, rather than their leaving office causing instability.