As far as priorities go, reliever depth is usually at the bottom of the list. If an organization wants to build a system with excellent relievers, it's usually by chance, or more likely that their good starting pitching prospects had to be converted to relievers. I mean, look at Andrew Miller and Dellin Betances.
The former was not a Yankees prospect, but both were highly touted starters who found infinitely more success in the bullpen. These guys pop up every so often, so the key is just trying to churn out as many arms as possible, and eventually you get some relievers who stick.
Another feature of this system creating relief prospects is that just about all of them are clustered at the upper-levels, save for the few players who were already converted to the closer role in college or early on in their professional career. Thankfully, the Yankees have a bevy of relief arms.
There's Johnny Barbato, who fans saw earlier on this season. I also wrote about him. He had a 5.54 ERA in limited time, but he has swing-and-miss stuff, so I think there's a good chance we see him again. The team also saw a bit of James Pazos last year, who threw five scoreless innings in the big leagues, and Tyler Webb should also be knocking on the door, as he's been pretty lights-out at Triple-A this year. Both of those guys are left-handers, so they could come in handy based on match-ups, especially when the team has an extended stretch without off days.
Nick Rumbelow and Branden Pinder were also on the Scranton Shuttle in 2015, but they won't be seen again for at least a year. They both just underwent Tommy John surgery. That's pretty devastating for a relief pitcher's value, so who knows when or in what capacity we'll see them again. Then there's Mark Montgomery and Diego Moreno. The former has been stellar and the latter has struggled, but again, these righties might be seen as deep roster candidates.
Beyond Triple-A, the depth gets a bit thinner, and there are only a few players that might have an immediate impact. Matt Tracy has seen time in the big leagues but has a 6.05 ERA in Double-A so he's still way off, and Jacob Lindgren is still making his way back from injuries as he recovers with High-A Tampa. He was the Yankees' top pick in the 2014 draft, and that draft essentially hinges on him becoming Miller-esque. His stuff is obviously fantastic, but if he doesn't make the majors this year, the chances of that happening dwindle significantly.
The next candidates for the Scranton shuttle include Jordan Foley, Matt Wotherspoon, and Jonathan Holder, who are all grinding it out in Double-A and High-A (Foley is the only one of those in High-A). Again, these guys are hit-or-miss, but given the Yankees' success with these arms percolating in Triple-A and as a 25th man, one of them might stick in the coming years.
As I've said, these players are not high-impact. The Yankees currently have the best bullpen in baseball and that really isn't changing any time soon. The biggest impact the fringes of the bullpen has is on off nights for the team's best relievers, and providing match ups when the team isn't rested. That being said, these things can change in instant.
With an injury or two Joe Girardi could suddenly be relying on these guys to make important outs late in the season. Considering their aptitude in developing these arms in recent arms, this part of the roster doesn't particularly worry me. And as far as the draft goes: just draft the best arms, particularly starters. If they end up becoming relievers, we won't know that for a few years.