Also, note I was using AirPort Utility on my Mac, not on my iPhone -- though you can probably do it with the iOS version, too. Of course, you'll need a data plan from your carrier.
First, in your iPhone Settings, make sure Wi-Fi, Cellular, and Personal Hotspot are all turned on. Now follow the directions shown in Settings under Personal Hotspot to join the network with your Mac over wi-fi. (In your iPhone Settings, you can change the wi-fi password, if you like.) The easiest way to switch wi-fi networks on the Mac is on the menu bar, so make sure your Network Preferences are set to "Show Wi-Fi status" there. Also, when you switch to the hotspot network, be sure to tell your Mac to "remember" the network, so you don't have to reenter the password next time.
Now take your AirPort device and reset it. On my AirPort Express, the procedure is to unplug it, plug it back in, and depress the Reset button with a paper clip end for about five seconds, till the amber light flashes rapidly. (To reach the button, I plugged the device in upside down.)
If all goes well, that should do it! After the AirPort device's restart, you should see the device on the AirPort Utility home screen with a nice green light beside it. Now you can easily switch your Mac back and forth between the hotspot and your primary wi-fi network -- and with the password, so can any other computer or mobile device within your newly extended range. To deal with the added battery drain on your iPhone, you can keep it plugged in to charge while the hotspot is active.
Update -- July 17, 2017
As I've discovered, things get a bit more complicated if you're trying to set this up while your regular wi-fi is down, if you don't have a data plan in place. That's because, without exising wi-fi, you can't use the T-Mobile app to get the plan!
If you get stuck this way, you can add funds to your account by calling *ADD and going through the automated system. Then call T-Mobile's customer service to buy the plan. (Or maybe the T-Mobile app will work if you turn on Wi-Fi Assist in the iPhone's Cellular settings -- but I haven't had a chance to try that yet.)