Working at a clinic is a great way to get experience, yes, but it's not necessary as long as you get experience somehow. All of my vet experience was through shadowing, unpaid internships, and volunteering and I got in.
So, if you can get a job and you like it, or if you need to get a paid job, go for it. But don't stress about needing to get a vet-related job, shadowing and volunteering work too (and can sometimes lead to jobs, actually).