The South seceded because a Republican was elected President. If a Democrat is elected in 1860, the fire eaters won't be able to convince people to secede. The crisis will be postponed until a Republican is elected.
The Republicans were an explicity anti-slavery party. Once one was elected President, a Republican could use the patronage system to begin building a Republican anti-slavery party in the South. While still limited in appeal, nevertheless you now have a political party in the south which will challenge slavery. This is far different from the Whigs who pursued similar economic policies and have anti-slavery members, yet had pro-slavery members as well. A southern Whig is acceptable. A southern Republican would not be.
John Tyler would not be objectionable to the South. He would probably favor them so there is no reason to secede. Presumably, political tensions continue to rise. If he is without a party, then the 1864 election might see multiple parties like the 1860 election IOTL. You'd have Tyler, a Republican, and one or more Democrats running based on their slavery positions.
Assuming a Republican wins on the basis of the North's electoral votes, then the Civil War breaks out in 1864 (if Tyler holds on, then it might be delayed until 1868 although political crisis would be terrible). The Deep South secedes first. The Federal government responds, and then the Upper South may secede on the basis of that response. The Republicans have 4 more years to build their party, possibly even in the old Whig strongholds in the Upper South. That plus the possibility that the fire eaters could do something to bring the blame of violence on them, might cause one or more states that seceded IOTL to remain in the Union (Tennessee is the most likely candidate).