If you have banking experience then First TN is HQ'd in Memphis. Just something to consider. Not saying there are any opportunities but something to look out for.
As far as a place to stays at ...I recommend Midtown, but I'm biased, I grew up in the middle of Central Gardens. So the only advice there is that there are no apartment complexes or buildings (with a couple of exceptions) you'll want to consider in that part of town. There are houses for rent and a bunch of upstairs garage apartments. No idea what the prices are like.
BTW, UT Memphis is Downtown, not Midtown. The dividing lines there are very clear...for example, you could live in a house on the North side of the tracks (in this scenario those 1 block south of Central Av.) and have gangs and cracks houses 100 yards away on the other side of the tracks but you would never have any problems. It's a weird place.