For years I've been asked, "Can I do voice-over part-time?" To which I confidently reply, "Sure... if you happen to know a producer who has enough work to keep you consistently busy, and who wants your voice on absolutely everything they do! (And if everything they do consists of national spots for major market campaigns.)"
Okay, barring that option, the fact remains "part-time" for any other business is 20 hours a week. So, it stands to reason if you were to dedicate ten hours a week to developing and working your skills, and another ten hours to promoting your demos--then you most certainly stand a far greater chance in becoming a full-time working voice talent. But that means you have to put in the hours.
Dedicate yourself to working on your career for no less than 20 hours a week, or part time, if you will. If you're only just starting out or re-branding yourself, positioning and prepping yourself for what the job requires, at the very least, 15 to 20 hours a week. This is the only way you will honestly stand a chance at becoming a professional. It takes commitment.
To add to this, it's vital that you persist in the face of a great deal of adversity. Not because you're a glutton for punishment, but because dedication and commitment to delivering a superior final product will offer you a track record before long, confidence in yourself, and will ultimately make you a very valuable asset to every production you grace. Working steadily takes a great deal more persistence than anyone might ever imagine, just like any other start-up business. So it's realistic to embrace this notion from the start, and from my experience, is necessary regardless of the business model.
Unfortunately, far too many talent coach once a year for only an hour or two, rarely (if ever) promote themselves, create cheap and often unusable promotional tools (demos and headshots), and barely put so much as an hour a week toward establishing a career for themselves. The work won't seek you out. And even if an opportunity were to come along--would you be ready to deliver your very best?
Realistically you should treat your would-be acting career as the small business it is. While you're expected to be skilled and agile enough to simply play with each audition and session, every other aspect of this business demands you approach it as a profession.