Handoff Not Working Between Apple Watch and iPhone

So, thanks to everyone's opinions on my Apple watch thread(as well as my own wants), I ended up getting an Apple watch series 6 yesterday. While I'm still playing around with it and figuring out where I'll use it most in my daily life besides the fitness features, I'm having a bit of an issue.
I've got handoff on on both the watch and phone. But when I try and compose a new email or message on the watch, then go over to the phone to try and continue it, the mail or messages apps don't show in the iPhone's app switcher.
Is there something else I'm missing? I've toggled handoff on and off on both devices, and still no go.
I'm loving this little thing so far. Seeing notifications and taking calls on one's wrist when away from the phone is neat. I'd just like to get handoff up and running. Thank goodness the phone call part of it seems to work!