Here is an HDPE 3/4" - 5/8" adapter for 44 cents at U.S. Plastics. Would this work for you?https://www.usplastic.com/catalog/images/products/tubing/62001p.jpg

The garden hose is unfortunately threaded so I'd need the 5/8 end to have threads. So weird how this adapter isn't available anywhere for the two most common standard sizes for garden hoses, as if no one on the planet is ever going to try and hook their old 5/8 standard diameter garden hose to the newer 3/4 standard spouts on newer houses.

Perhaps U.S. Plastics has another adapter I didn't see. They have a vast selection of these handy tee, connectors, adapters, etc. If you absolutely can't find an "official" adapter, can you splice the 2 hoses together with one of those plastic hose repair cuffs? In a pinch for my soaker hose system, I also tucked a hose with a 5/8" outer diameter into a hose with a 5/8" inner diameter, with hose clamps at either end to keep it tightly together.