Everything you do will increase your chances of getting a better job but nothing will guarantee it.
What sort of job are you looking for and what's your degree?
Important to keep in mind that AAT and ACA/ ACCA are two different types of qualifications. ACA and ACCA offer chartered status and is used by all the largest accountants (along with CA); you are either qualififed or partially qualified. AAT is more common within smaller accountancy firms and has multiple levels that can be completed.
I am studying AAT, and I have back-up (my dad's firm). I am getting all the practical exposure I need from my family firm. However, I think, jobs and studying are two different things. You can study for accounting while holding a job in sales. These days, volunteering is also counted as experience.
Increase the chances of getting employed in accounting by being a professional certified accountant. If you are new to accounting, then start off with AAT and progress on to either ACCA or CIMA. One advice I have always given to anyone starting off an alternative career is: never let go off of you current job till you are a certified professional. This will keep your income inflow stable you can also learn at the same time.
Im studying for the C FAB qualification as a non-numerical undergrad. I think it will show commitment to career as a competency as well as possibly leading to exemptions. I spoke to a partner who said that one of their trainee goes on many science companies they audit as she has a bioscience degree.