First, as to the question of wide-spread "sympathy secession" movements as far as the Old Northwest, I think that robertp6165 has a fair point: there is going to be a lot of uncertainty about Jackson's use of force. However, the Northwest in particular supported the Tariff regime inaugurated by the Tariff of 1824 (eg. Van Buren used the Tariff of 1828 in part to get PA to support Jackson for the Presidency). Hence, I don't see Jackson precipitating widespread secession in Illinois or Ohio, but I do think you might see perhaps some variant of the Kentucky and Virginia Resolves: statments of dissent against the national government. At the most, you might see non-compliance with Federal orders (this has the potential to be pretty chaotic, since short of military compulsion the Federal government of the 1830s has very little ability to coerce states to comply with its orders).
Secondly, given how close the crisis got to confrontation, if South Carolina actually does act to formally secede, Jackson will act very fast to counter-act it, much faster than Buchanan did under very different circumstances. Of course, there could be some problem in getting authorization to act from Congress: OTL Congress agreed to Jackson's Force Bill in concert with the Compromise Tariff. If Jackson is put up against the wall, Henry Clay may use the chance to win some greater support for the American Plan from Jackson as a price. Because of the very dicey political situation, however, I think a united opposition to Jackson won't easily coalesce. Hence, I doubt you'd instantly see an alt-Confederacy spring into being, motivated simply by the issue of secession. TTL the issue is almost strictly the constitutional / legal one, rather than the deeper, cultural question of slavery. In 1861, it was much easier to feel that States Rights' was also a matter of Southern Culture. In 1833, much less so. Indeed, Jackson himself was an impassioned defender of both states rights (for example, in regards to internal improvements) and of slavery; he was also an ardent Unionist.
IMO the most likely result of a greater confronation with an explicit secession is a potential resolution of the legality of secession under the Constitution. I don't mean the issued is settled with the weight or finality that it was after OTL's ACW, but simply that it may well provoke a more formal elucidation of an official legal position for the Union in future. For example, a Constitutional ammendment similar to that adopted by the *USA in DoD: one that permits secession, but by outlining a means to permit it, makes it very, very hard to pull off (3/4s vote of the state legislature to petition Congress, a 2/3 or 3/4 vote of both Houses of Congress, followed by a convention to ratify the petition of the state legislature, for example). The importance here is less that the ACW would errupt in 1833, but more that the nature of the slavery debate will be different. If the South knows its right to secede is guaranteed, it may well be willing to coutenance a Lincoln-esque Free Soil Administration; at the very least, it may make a compromise more likely that would prevent a real ACW - style conflict. If the South secedes in such a manner that contradicts the Constitutional formula, than the North / Union may be presented with a clearer call to action to stop it.
Additionally, such a crisis would have profound effects on Jackson's 2nd term and on the Democrats for the next few years, particularly if he concedes some support for internal improvements to quiet the Old Northwest, for example. Perhaps Jackson's actions will provoke a more powerful Whig Party as an opposition and so greatly impact later political contests: in 1840 and 1844, the Whigs were fairly disorganized; 1844 was a particularly bitter pill since John Tyler's ascension to the Presidency proved to pit the Whig Party against itself.