Well first it is important to understand that Breckenridge's strength in the 1860 election in the South is often misunderstood. The Lower South was very much a three way race with Douglas, Bell, and Breckenridge all capturing large portions of the vote. Breckenridge in fact won only slightly more than fifty percent of the vote in the seven states that would become the CSA, and he would have lost several states if Douglas's and Bell's votes are counted together (as both stood as Unionist candidates). In the Upper South Breckenridge's position is worse, with Lincoln surprisingly winning a large number of votes in a few border states, and overall Unionist candidates performing better. More to your original question, Breckenridge like most in the Upper South advocated for compromise after the initial seven states left the Union and didn't join the CSA until well after Jefferson Davis was elected president.