Also I go outside without a phone on occasion. That feels like you leave a burden behind, and you are somehow more free. The phone is that easy thing you just grab to do a quick check of something on Wikipedia, or you happen to notice a notification. It is a distraction-device, keeping you busy. And among strangers, feeling less comfortable, well you can grab your phone and start staring at it. This behavior is like with smoking. Just like the relaxed cigarette cowboy in the ads, but now you casually light up the screen and be cool.
It has WiFi but no cell phone. Really looking forward to just being out for runs, with no possibility of contact or checking notifications etc.
[0] I know I still have IMs, but mails tend to be things I need to do at a computer and seeing them only stresses me with things to do for later.
Something I have done (accidentally at first, now on purpose) is to not respond to messages (personal) quickly, most of the time. People adjust to that rather fast and stop worrying so much. People in my circles know now that I am rarely going to answer within a few hours and expectations are adjusted. So then going outside for a few hours with no phone is no longer a "thing" - you just do it and people will expect you to get back to them when you do.