I see, Overcast provides a possible solution then.

If you pay for the Overcast Premium subscription (£8.99 a year) it gives you the extra facility to upload podcasts to their service and then play them back in the app. It is limited to 2GB of storage, but it might work for you.

When you've uploaded them they appear in an "Uploads" playlist in the app.
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Remind me to change my signature to something more interesting someday