Two name options - Either Five Mile Hideaway on the Trace or The Hideaway on the Trace
The name "Five Mile Hideaway" or "The Hideaway" should be big and "on the Trace" should be small underneath.

This logo needs to go on a styrofoam cup and be created from medal so it doesn't need to be to color intense. So black only, brown only or dark green only

The creek is what is five miles so they like the creek and a deer in it . The trees along the Natchez trace look like this below. Icons could be creek, deer or I really like how this Duck Hunter logo included is subtle but strong.

- https://www.google.com/search?sca_esv=62c9ed8991bec703&sca_upv=1&q=natchez+trace&tbm=isch&source=lnms&fbs=AEQNm0DrTR52rgbtMpR539hbBsEHk7I0rRN9NqJUdEX_lxJRCz0dXbs4i2HSoniqRVWj5VlLyhcWxKhp3eZ1U5haDqEV2xb6OzDT-lK3vANdZvYBjkd0Lph516nhXGYYR1RrRk3Kg1I32FKdxEzVFd35I-EV6PM5jGKXVYAiics5fk_GEv5M5ZKVgSI7O1X_s9Kq0fyc8g0Mw9FVc_EUr2AtIaAARKta-A&sa=X&ved=2ahUKEwjkhqyx6vGGAxVMG9AFHUILAXUQ0pQJegQIHRAB&biw=2091&bih=1074&dpr=1.6


https://www.shutterstock.com/image-vector/set-vector-adventure-mountain-outdoor-vintage-1930316537

https://stock.adobe.com/search?filters%5Bcontent_type%3Aphoto%5D=1&filters%5Bcontent_type%3Aillustration%5D=1&filters%5Bcontent_type%3Azip_vector%5D=1&filters%5Bcontent_type%3Avideo%5D=1&filters%5Bcontent_type%3Atemplate%5D=1&filters%5Bcontent_type%3A3d%5D=1&filters%5Bcontent_type%3Aimage%5D=1&filters%5Bcontent_type%3Aaudio%5D=0&filters%5Binclude_stock_enterprise%5D=0&filters%5Bis_editorial%5D=0&filters%5Bfree_collection%5D=0&k=deer+creek+logo&order=relevance&safe_search=1&limit=100&search_page=1&search_type=usertyped&acp=&aco=deer+creek+logo&get_facets=0&asset_id=255130547

https://stock.adobe.com/search?filters%5Bcontent_type%3Aphoto%5D=1&filters%5Bcontent_type%3Aillustration%5D=1&filters%5Bcontent_type%3Azip_vector%5D=1&filters%5Bcontent_type%3Avideo%5D=1&filters%5Bcontent_type%3Atemplate%5D=1&filters%5Bcontent_type%3A3d%5D=1&filters%5Bcontent_type%3Aimage%5D=1&filters%5Bcontent_type%3Aaudio%5D=0&filters%5Binclude_stock_enterprise%5D=0&filters%5Bis_editorial%5D=0&filters%5Bfree_collection%5D=0&k=deer+creek+logo&order=relevance&safe_search=1&limit=100&search_page=1&search_type=usertyped&acp=&aco=deer+creek+logo&get_facets=0&asset_id=802557200

Five Mile Hideaway or The Hideaway