I quietly watch them teach several students. Most of all, I want to see the students improve during the lesson. If the coach doesn't can't find a way to help, that is a bad sign. Sometimes I have also asked for advice from other skaters, but learning is such an individual thing, that that doesn't help much. One thing very definite - I would never use a coach who seemed in any way abusive. Yelling or getting mad at a student or another skater is seldom called for.
But after that, there is no substitute for trying a lesson.
If most of your experience was as a child, you may find your needs are different. You may be newly afraid of falling (get over it - by practice). Things you used to learn by watching may now need words too (very common). A technique that requires more flexibility than you can realistically attain may not work for you - I picked one coach because she looked super-graceful, but I later realized her techniques couldn't possibly work for me.
I personally do need a coach who can demo what I want to learn. Of course you can't tell whether they are doing everything right, but things that look good to the eye often are. For example, the coach should be able to skate with speed, power and grace. When not pushing speed to the limits, or doing jumps, their skates should be almost silent on the ice. My coaches must also be able to explain things in words - I love physics-style explanations. But that's just me.
Another thing you might possibly look for - if he/she will be choreographing routines, decide whether you like the routines he/she has choreographed for others. Is it interesting? If you have a strong sense of rhythm, do his/her routines employ music with a strong beat?
Then there is lesson price, convenience and available hours. Both of the coach, and of the rink(s) where the coach teaches. (Of course, it is not always a good sign if the coach is TOO available. Some of the best are hard to get.)
BTW, is one of your old coaches still available, in your area? Consider that.
Some (most?) of the coaches with elite level students show favoritism, giving most of their time to their most competitive students - especially before competitions. Decide whether you can live with that.
If you might want to do Ice Dance, you pretty much need a coach of the opposite gender. And then you also may want someone with the most common style, so you can skate with others. For example, since you are in the U.S., you probably want someone who, when the two skaters are facing each other in closed (Waltz) hold is usually directly opposite the other skater, and looks directly into the other skater's eyes. Yes, you can get closer body positions by offsetting the hold, but that is not really American standard, and is actually against what the USFS says is ideal in the rulebook. Also, you want a coach who can adapt to dancing with less skilled partners, and not making them look clumsy.
Another thing in ice dance or maybe pairs (I haven't done pairs) that might or might not matter is a skating style that roughly matches yours in certain respects. Specifically, some skaters lean and balance blades into the motion (lean forwards in forward skating, backwards in back - e.g., Davis and White), some lean and balance away from it (e.g., Virtue and Moir). Some turn their torso into and before turns, some counter-rotate. A coach can probably adapt to the latter, but the former may be too hard for them to change. (OTOH, Davis and White, and Virtue and Moir, had the same coaches at the end, and were quite successful, yet the two couples had very different dance styles.)
Finally, a lot of Dance and Pairs skaters have trouble finding partners. It is partly the coach's job to help - ask their students whether they succeed. And ask the coach whether they have rules about who their students can dance with, if you care. Many coaches say you can't dance with anyone they don't approve of first.
But really, you have to decide what you want in a coach.
I wouldn't ask a figure skating director, or someone at the front desk. They are not supposed to show favoritism by advocating someone. If I remember right, that was a PSA rule.
Good luck!