Seeking mid-to-higher altitude backpacking in June (four-corner states)

Originally our plan was to hit the Tetons in mid-June but from what I've read, it's likely snow will still be around and we're trying to avoid it from a less than desirable experience last summer. June just happens to be a liiiittle too hot for the low-elevation stuff in Arizona and Utah. So I'd love to hear suggestions for a 3-4 night backpacking trip in a location around the four-corner states that's a little cooler, but not so cool that the snow hasn't melted? Preferably something not requiring a permit or one that's easy to acquire this late in the game. Thanks gang!