Pokemon Go, cannot connect at all outside of wifi (not using Pokemon Trainer)?

Problem started just a couple of days ago. iPhone 6, fully updated. I have wifi in my house and have no problems connecting and capturing the [very] few spawns. But as soon as I leave my house, it stops working, even in areas where cellular service is at 4 or 5 bars. I can't use Pokestops, see spawns, or hatch eggs.

For example, across the road from my house is a Pokestop. Until a couple of days ago, I would pass by on my walk and access the Pokestop along the way. Now, I walk across the road and approach, and I see the Pokestop but the picture won't come up and a spin produces nothing. So I exit out and restart the game, only to see the (by now familiar) message that I can't connect and try again to log in. This is using my email, not Pokemon Trainer, and the server page on the website says that all is well, green all the way. I try many times, but cannot connect at all, until I'm back in my house with wifi.

I went to an area rich in spawns and Pokestops just now to test it. Sure enough, nothing comes up, cannot even see distant Pokestops or Gyms. Restart and am told I cannot connect. This is frustrating. I'm nearly out of Pokeballs, have only two eggs left (which won't hatch because the game isn't recording my walks anymore), and cannot play because my house spawns *maybe* two Pokemon a day, and they're pretty common little guys.

Does anyone know how to fix this problem? It seems to have started after (I think) the game updated, as I no longer see a battery saver option.