You're not out of luck, but you'll need to be persistent! You have two options: walk-up permits or cancellations.

I got a walk-up permit from Happy Isles in 2016. I had actually reserved a permit from an alternative trailhead and was planning to meet up with the JMT farther down the trail, but I checked for a walk-up when I arrived and I got lucky. This is a fairly risky option, especially if you don't have a backup plan.

The other option, like other posters mentioned, is to pick up someone else's canceled permit. My dad was able to get a permit this way last year after losing the lottery. It requires some persistence -- canceled permits will show up on the permit availability report, so you need to open the permit website every day and refresh it, hoping to see an available permit for your trailhead and date. It looks like the application has moved to this year so there might be some differences, but I imagine the overall process is pretty much the same. If you're persistent and a little flexible with dates you'll almost definitely be able get a permit this way... but you'll have to be very persistent. (I actually quite like the JMT permit process for this reason. If someone?really wants a permit there's a way to get one even after losing the lottery.)

One more thing -- I'd consider starting from Lyell Canyon as well. Obviously take a Happy Isles permit if you can get one, but you'll increase your chances if you consider both trailheads. If you get Lyell Canyon, I'd recommend getting to Yosemite a couple days early and then you can do some of the Happy Isles section separately -- for example, the iconic first few miles from Happy Isles and Cathedral Lakes (from Tuolumne Meadows) can be done as day hikes, and Clouds Rest can be done as a long day hike or a one-night trip from a less competitive trailhead. This way you still get to experience the entire JMT, plus you get the benefit of doing it with a lighter pack. That said, even if you miss the Happy Isles to Tuolumne Meadows section entirely, the rest of the JMT obviously still worth it!