Volunteering can't hurt. It makes you known to them, does give you a chance to network, and gives you something to put on the resume that looks a lot better than "foster dog trainer."
But remember that when you're volunteering, you're there to cheerfully help however you can. If you spend all of your time trying to schmooze a job instead of doing your volunteer duties, they will see through you and fast.
I volunteered as a new grad...not where I ended up getting hired, but at a community clinic. It definitely made a favorable impression on interviewers.
As far as job hunting...well, you've said that moving is not an option. Fair enough, but that means you'll likely have to compromise in other ways, as in what you will do for this first job. You may have to suck up a commute that is longer than you like, or work hours you're not wild about, or even work a speciality or a setting that you don't care for. But keep in mind that this is your first job, not your forever job. The goal of your first job is to gain that magic year (or two) of experience so you can move on to something more to your liking. Think of it as paying your dues.
You don't want to be too picky about your first job...because wait too long, and you become an old new grad. And old new grads have even worse job prospects than new grads. For one, they've aged themselves out of a lot of new graduate residencies or new grad positions that require you to have graduated within the last 6-12 months.
Best of luck.