Podcast episodes don't show on iPhone when refreshing list?

Normally I download all episodes in iTunes and then sync them onto my iPhone. When I go on road trips, and I'm away from my home computer, I have been unable to get any podcast to show the latest episodes. I was under the impression that I didn't always have to sync with my home computer to get episodes and that I should be able to download the newest ones anywhere I have internet access. Perhaps I was mistaken?

In the settings of my Podcast app, I have the "Subscribe" feature turned on. When I swipe down on the podcast that I want to update, the animated icon shows that it's "working", but then no new episodes show up. Yes, there are new episodes because the podcasts I subscribe to have a new one each day.

What am I doing wrong? I want to download episodes when I'm away from home.

Thanks!