Going into the 1st Confederate Congress/Convention, 3 people really had a shot for the (provisional) presidency, Jefferson Davis, Robert Toombs, and Alexander Stephens.
Many people liked Toombs, but he had a horrendous drinking problem, and his conduct at one of the many social functions in Montgomery at the time destroyed his chances for the presidency considerably. This event didn't completely destroy his chances, so for one of the first changes, let's have Toombs drinking problems be witnessed by more/become apparent at a few more functions. Let's say that these are enough to prevent him from any serious consideration.
Now, the second step of this is to try to get the Georgia delegation behind Stephens. This needs to happen early on in the Congress, as by the time the Georgians decided to sit down and decide on their candidate, the united Mississippi delegation had got enough support for Davis. Let's say that after Toombs' drinking problem comes out, the Georgians sit down to decide on who they want to nominate. Howell Cobb disuades any thoughts of him being nominated early on, and Toombs, close friends with Stephens, supports Stephens to be Georgia's candidate seeing that his own chances are gone.
Now, in OTL, Stephens told many that he didn't want the presidency even though he had a lot of support. This dispersed most of his support, so let's have Stephens not explicitly say he wants the presidency, but he doesn't outright deny it either, keeping him in the running. Combined with Toombs support during the Georgia discussion, the other less prominent Georgia delegates rally behind Stephens, and by the end of the talk, the Cobbs support a Stephens presidency.
In OTL, as the presidential election in Congress came closer, Tom Cobb went around and told the other state delegations that Georgia had decided on Howell Cobb as their nominee. This destroyed a lot of good will towards Georgia, as many saw Georgia trying to push their candidate on the others. In addition, the Georgians hadn't decided on a candidate at that point, and these talks destroyed Georgia's chances. We have avoided this in this scenario with Georgia deciding on it's candidate early on.
Now, we need to build up Stephens' support in the other state delegations. Louisiana is probably Stephens' greatest supporters outside of Georgia. Many of Louisiana delegates liked him, and with Georgia rallying behind Stephens, Louisiana will probably throw it's weight behind Stephens. Mississippi is united behind Davis, so we can write them off. South Carolina is divided, but if we can get them behind Stephens, Florida will follow SC, and then we have four states, enough for a majority. (Each state delegation has one vote.)
Lawrence Keitt of South Carolina approched Stephens in OTL and asked directly if he would accept the presidency. Stephens was indecisive, and Keitt came back from the discussion disheartened. Keitt will probably still approach Stephens ITTL, and Stephens will be much more supportive of Keitt's efforts to unite SC. Many in the SC delegation were polarized against Davis, and many of the lukewarm Stephens supporters in OTL who went over to Davis will remain in the Stephens camp. While South Carolina will probably not be unanimous, Stephens will have the majority, and with it Florida.
The other interesting aspect that could appear with a Stephens presidency is the possibility of Jefferson Davis being the Confederate general-in-chief. Many supported him for this position in OTL with his War Department experience and his service in the Mexican War, but his securing of the presidency put these calls to rest. With a Stephens presidency, I think it is highly likely that Davis could at least secure a very high rank in the Confederate Army. The War Department could also be a possibility, but I am not as sure on that position.
I am unsure exactly how Stephens would govern the Confederacy, but does anyone have thoughts on this scenario and the aftereffects?