Depending on how the LSU-Bama-Ole Miss-A&M quadrangle plays out, our Vols will be anywhere from 4th to 6th in the SEC's pecking order come bowl season.
At best, 3 SEC teams get into the New Year's Six bowls (more likely, just 2) and we go to the Citrus as the #4. We would then be matched up against Michigan, Penn State, Wisconsin, or Northwestern, whichever of them comes in third behind Ohio State and Michigan State. If Iowa is the #3 B10 team, that poses a problem for the Citrus Bowl: their contract gives them rights to the best SEC and B10 teams not in the New Years' Six CFP bowls, but don't think they'd want a rematch of last year's TaxSlayer Bowl. In that case, they'd probably agree with the two conferences to juggle somehow. I personally think Tennessee - Michigan is the most marquee matchup we could reasonably hope for.
More likely, though, we'll be one of the SEC teams in the #4-#9 pool, where the SEC gets to assign its teams to the TaxSlayer, Liberty, Outback, Music City, Belk, and Texas Bowls. They're all officially considered co-equals in prestige, though some have significantly more history and tradition than others. I listed them in the order of their lineage: TaxSlayer (Gator) goes back to 1946, with Liberty closest behind from 1959. Outback (Hall of Fame) dates to 1986. So as far as the bowls themselves, getting into the Liberty or Outback bowl would probably be our best "marquee" bet, because a repeat trip to the TaxSlayer is very unlikely for marketing reasons.
But I don't think Tennessee wants to go to the Liberty or Music City bowls, ever, unless they are desperate for a bowl home. Getting out of the state is both better for recruiting exposure, and more of a treat and reward to the team. Even if it's just going next door to North Carolina for the Belk Bowl, I think the administration prefers that to a bowl in Nashville or Memphis. So Florida (Citrus, Outback), Texas (Texas) or North Carolina (Belk) seem most likely.
If it's the Outback Bowl, the matchups look a lot like the Citrus, already discussed. One of those also-ran B10 teams would likely be our opponent. Again, a Tennessee - Michigan matchup seems most prestigious of the realistic options.
If we go to the Texas Bowl, we could get a game with Oklahoma State. Meh.... Another possibility is Oklahoma, but I don't think the sponsors would want a Tennessee - Oklahoma rematch. There don't seem any other likely dance partners for us out of that conference.
Finally, we could meet up with Florida State, Duke, Pittsburgh, or North Carolina in Charlotte at the Belk Bowl. After the Citrus and Outback possibilities vs Michigan, these are my favorite possibilities. Any one of those four would be a fun game. We don't play the ACC this year in the regular season, so it'd be a new conference for us (with a nice lead-in to next year's Va Tech matchup). After a Michigan date, I think any one of these four teams in Charlotte would be our next best option.
TL;DR: facing Michigan in the Citrus or Outback Bowl would be our most 'marquee' realistic option; otherwise, getting one of the upper-half ACC schools (FSU, UNC, Pitt, Duke) in the Belk Bowl seems next best.
Go Vols!