my old non smartphone had this ability...and it worked great.

Here's what I'm looking for and why-
The other day I went to a buddy's house after work. He has no cell service at his home and many places around me don't. I was there about 5 hours. In that time my phone went totally dead because it was constantly trying to get a signal. My old phone would look for 5 minutes and then would go in to power save mode and stop searching UNTIL I turned the screen on. Then it would search again...screen goes back off and 5 minutes later it was back in power save mode. Power save mode basically made it stop searching for a signal until I touched the phone....That wasn't exactly it would still search on its own like every half hour instead of constantly searching.