A lot of companies will ask and the only reason they want to know IMO is HR has rules as to how much someone is worth based on their prior pay rate. My wife had this happen and it cost her thousands (like 10k) on a job offer because she was 100% honest with their HR department who does salary negotiations. Her new boss was very okay with her salary request but HR killed it. I know her boss and she was pissed with HR as she'd been an advocate for my wife at the company she was coming from.
They can ask and will ask in many cases but your employer/former employer will not give that info out. I know there are ways they can get it but it'd be shifty enough that I wouldn't want to work there if they did it (through employment verification services where lenders can request your salary).

Now you're in a job and it's likely common knowledge what a phlebotomist makes. Know your worth and only answer directly if asked, do not volunteer your current salary. If your current job is a union job, know the scale as the new employer could easily look it up using your start date there and know where you're at in the grade level steps.

Good luck