He's probably running in Massachusetts, where he might conceivably be elected on Lincoln's coattails. Garrison was a determined abolitionist, but I suppose that he might make the decision to act as a political leader rather than a organizational leader. It means that Garrison, previous rhetoric to the winds, must as least publicly reconcile with Abraham Lincoln.
This is something he would not do OTL, but might do with the same PoD that would cause him to enter politics. Garrison OTL was a very sensitive and very determined figure generally unable to compromise, this PoD would involve him recognizing that he could do more good in congress even if he had to bend a bit more to do it.
Garrison goes from lukewarm supporter to staunch ally after the Emancipation Proclamation, and while Garrison would really rather free all the slaves immediately, Lincoln is able to prevail on him the fine points of the Emancipation Proclamation.
In the end, Garrison's own political role is overshadowed by Lincoln. Its during the Impeachment of Andrew Johnson that Garrison changes history. A barely compromising figure of moral determination, Garrison only reluctantly supported Johnson as Lincoln's VP. And when Lincoln died in the hands of a Pro-Confederate assassin, Johnson's obstruction of Reconstructionist policy in the South made Garrisons blood boil.
The Powerful speech against Johnson, a determination to make reconstruction work and a massive change in the history books would lead to President Ben Wade, the first un-elected President of the United States. Wade would order Federal troops to arrest KKK members and ordered massive trials of some of the CSA's worst offenders.
Garrison, retiring in politics after a second term in 1872 (to tend to his stricken wife), would continue to support reforms until his dying day. Reconstruction would not be halted or aborted for another 40 years, at which point Black political leaders had emerged at the highest levels of the state government. While Booker T. Washington would ultimately be defeated as the Republican Candidate in 1892, the fact that he had won the Republican Nomination showed just how far and how fast Garrison's vision had gone.