There was a split in the conspiracy
Hauser, Lucy and Beauregard were on one side of it and whoever is releasing the prisoners on the other.
Tommy Madsen is one of the good guys.
If the 63's are being sent to track him down then logically that means that he's an enemy of whoever is releasing the 63's. Add in the fact that there's some doubt about whether or not he actually did
the crime that got him sent to Alcatraz in the first place, and there's a strong case for Tommy fitting in somewhere on the sliding scales of Anti-Hero
- Are we ever told what Rebecca and her partner were chasing him for? He was a "suspect," but for what crime? Perhaps it's a setup, or assumed suspicion, or perhaps he did something that was necessary to stop The Plan. That would bolster this theory.
- Pretty much Jossed in the series finale. Where he stabbed Rebecca critically and with little remorse and thanks to the cancellation. He pretty much ended up murdering her.
Barclay Finn is integral to the conspiracy
Barclay Finn (who Sylvane home-invades, then kills) is integral to the conspiracy. Hauser tries (and seemingly succeeds) to get Rebecca to forget about him. But he had one of the keys, and was specifically targeted by "they", to such an importance that grocery store petty thief Sylvane upgrades himself to cop killer in order to get to him. Why did Finn have one of the keys? Why did "they" want him killed?
Jump or whatever it's called was arranged by time traveler from 2010s-2020s and used Alcatraz as a mean to return.
He jumped with them and will appear last, right now he arranges murders of people who would interfere with his future self. He doesn't target Hauser's group because he doesn't know that in new timeline he will be his foe.
They somehow used whatever was in Tommy Madsen's blood to jump the prisoners.
Tommy Madsen's blood is being taken for no reason, and he's always in the hospital despite being perfectly healthy. He had no idea why.
Tommy Madsen will be the Big bad of Season One
He seems linked to all the criminals so far. (as of episode seven)
The Warden is responsible for everything that happened
He found some sort of alien entity under the prison and under its direction he started all those experiments on the prisoners. However, after a while he could not keep it all secret from his staff and they started to ask questions and demand he bring them in on the secret. These probably included people like the Deputy Warden, Lucy and Dr. Bauregard. when they got too close, he got the entity to teleport everyone in the prison at the time into another dimension or store them all in suspended animation.
- Tiller wasn't affected, though, and he aged normally before Jack killed him.
- Tiller was absent from the prison when it happened. He would have been the only one with the authority to stop what the Warden was doing so the Warden had him off the island on some pretext.
Tommy Madsen was the reason for many parts of the series thus far
Beauregard and Lucy are both sympathetic to Tommy because of his treatment - Lucy wants to help, Beauregard doesn't know what the blood is all about and while he may be sadistic, he doesn't seem to actually want to hurt the prisoners. (well, there was Clarence, but he's not as manipulative as the Warden or E.B. - Beauregard still has a soul) Tommy Madsen was an accomplice to Lucy and Dr. Beauregard, probably helping them get into the time machine, but Tommy was captured before he could join them. Lucy and the Doc arrive at around 2000 or slightly earlier, they then contact Hauser. Hauser thus knows:
1) The 63s are coming back
2) Tommy Madsen is someone who they may be able to have an ally
Hauser also has actual 63s to show to the FBI officials who are empowering his task force, verifiable people who made the jump, but he needs more.
Hauser attempts to recruit Ray as a way to approach Tommy rather than as an evil cop or prison official - blood being thicker than water. Ray turns him down and raises Rebecca instead. Ray thus isn't overly surprised when Tommy shows up and actively hides him. Hauser needs Rebecca not because she's an able detective, but because she has a foothold on Tommy's psyche.
This explains the importance of the Madsens to Hauser as well as why he would know that the 63s are returning and thus able to marshal his forces.
As for the blood - I'd suggest that perhaps Tommy's being used simply as a universal donor supply (say type o-negative or such), they're preparing the blood with the colloidal silver to transfuse into the other prisoners for their various time jumps. (though why that needs to be done as early as three years before the mass exodus is still up in the air, perhaps they didn't all disappear in 1963, but one at a time until said mass exodus occurred.