Bottom line: July and August are the most popular times, for good reason. The trail conditions and weather are most favorable then.
First off, people through-hike the Enchantments year-round, including on skis in January.
For most hikers, who are looking for a summer hiking experience, the best time to go is when the snow has melted off the trails but before the weather turns sour in the fall.
Starting around July 4 of a typical season, the trail is mostly snow-free and the lakes have melted out. The weather in the Pacific Northwest famously turns nice for 60 days starting July 5. Usually.
The summer days are longer so you can start earlier in the morning and have more daylight to finish your hike.
Late August brings occasional thunder storms. The wide-open, exposed Core is no place to be in a lightning storm. Those storms, unfortunately, bring wildfires to the region. And that means forest fire smoke.
By Labor Day (first weekend in September), the blue skies begin to intermittently turn gray. Rains return, lightly at first. Dry weather is much less reliable. Snow is possible.
As of the beginning of October, it’s officially the season of potential snowfall in the Core. Serious snowfall can happen earlier. Regardless, it’s something that can make misery of a nice trip if you’re not prepared.
I’ve hiked the snow-covered Core and it’s a beautiful, unique landscape. The lakes are partially or completely covered in ice. The peaks hold large snow fields. The trail is a deep rut in the snow.
Snow also adds time, complexity and risk to the journey. You’ll want to have the right hiking shoes and be skilled and equipped for travel over snow and ice.
But I know people who have made the trip on backcountry skis and said it was unforgettable.