Watcher hit the nail on the head with "In the end, Jet just wants somebody about whom he can care [for], and who will let him care." As much as Alysa is played off to be different than Spike and gang, Jet is still essentially beholden to her as he is to them.
The truth is he can not be the same kind of Black Dog that he was when he was with the ISSP. Fad is the one who seems to be controlling the chase tactics as he and Jet pursue Udai for the first time, or at least it seems that way since Fad is the one who offers to head around front to trap the assassin. To me, that isn't so much of Fad having charge over Jet as it is Fad trying to facilitate the chase for Jet. Throughout the series, Jet's reputation is that of the "once he bites, he never let's go," but that wouldn't always be possible if there wasn't someone like Fad by his side to help corner the criminal. Police units are not unlike military units in that a team has to function in sync in order for the perp to be apprehended; one man can not do it all by himself. Jet tells Spike that he lost his arm by being too headstrong, which is basically the same thing as admitting that he didn't think about what would happen before he started the chase. Why would he? He had a partner to do some of that for him.
Of course, Fad's help was only on the more immediate level. In addition to him, Jet had the ISSP brass to tell him what crime to investigate. It stands to reason that Jet and Fad would not always be the only two detectives gathering evidence on a case, especially one involving an assassin, so information would arrive second hand. It isn's shown, but no officer goes into a pursuit without another unit (or units) to back him and his partner up. With the police, Jet has an army of support, but, once he leaves, those back-up units ae gone to him.
After the ISSP, Jet is now responsible for getting his own information. As the show progresses, Jet runs into situations where there is a moral dilemma attached with the pursuit. Yes, he does break out the nickname for his pursuit of Alysa's boyfriend, but he already knows that the guy is likely to be found innocent before the chase begins. Later on, when he leaves the chessmaster alone, it is just as much for the old man as it is for Ed; the fact that the gate corporation's greed was a contributing factor to the accident is not lost on Jet. In both cases, something takes precedence over Jet's task of pursuit, and this is not what he would have had to dealt with in the ISSP: he has to determine if he's justified in his pursuit before he can make the "kill." That's what I meant when I said that Jet could no longer pursue with impunity.
That's also why he needs Spike and Faye as support. Why was it so important for Spike to accompany Jet in his search for Faye in JJ p. II? That's because he needs someone to back him up. I don't think there's a better example of how Spike and Jet work together than in Asteroid Blues. While Spike is in pursuit of Asimov, Jet is relaying the pertinent information to him over the com and coming to his aid when the syndicate goons start to interfere. There is synergy when two men function as unit, taking on individual tasks in support of the whole, and Jet isn't a mercenary, he's a converted cop. It wouldn't make sense for him to suddenly go against all of his training when he adapts to his new way of life. Although he is certainly strong enough to have survived a little bit before he met up with Spike, I still think that he would not have lasted this long as a bounty hunter.
Lastly, what did you make of Jet expression as he pursued Alysa and her boyfriend (I'm too lazy right now to look up his name)? I'm not sure what I would call it, but, instead of determination, I saw what looked more like resign with a little bit of fatigue / weariness. Compare it to the face Jet has when he is pursuing Udai the first time. That's where I got the part about him not identifying with his nickname anymore. A dog is trained not to question its master, and for that reason dogs show a kind of determination in their pursuit that is not easily duplicated in a "free-thinking" human adult. I would say that Jet's expression in Ganymeade Elegy indicates just the opposite of determination, resignation.