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Some people (especially people with smaller chests, like yours) use KT tape to bind. You can pick it up from any nearby sporting store, or grab it off Amazon. Costs around $12. Youtube has some videos on what to do with it.
Added bonus if you're staying with or close to family, if anyone finds it in your stuff, it's KT tape. It's used for sport. So you can just say you noticed some pain when jogging (or decided to try jogging) and bought it to help.
Would either of these suit your needs? If yes, put the one(s) that would work on an Amazon wishlist and post it. I will buy it for you.