If you’re on your employers network, they can still see you regardless of any VPN. What I should say is, it could be done. I don’t know how complex the network and software running it is. Maybe you know for a fact that they track you via their VPN service but there are other ways. Router for instance.
Their local network(wifi) is the first thing you go through, then onto the internet network(world wide web via a network of domain name servers) then then on to a VPN network(servers) and then to domain name servers again to your final destination(a single server such as the one freerebuplic resides on).
Your best bet would be using a phone and your data plan/cell signal(4g/5g) to get on the web instead of going through their wifi for personal stuff or just do it after hours.
I have no idea how you could configure two VPNs. The goal would be for the only traffic they see from you is going to your VPN and not the final destination.
Lot of effort to go through to do personal stuff at work, uhm, when you should be working.
In theory, if I connected to my VPN, and then connected to my employer's VPN, then they would see my VPN provider's server as what they would be connected to. In other words, they would think that my VPN provider's server was my computer, and they would see only the traffic intended to go to them. That was my theory, anyway.
I work from home, so I'm not connected to their WiFi. I'm connected to my WiFi, and load their VPN software when I need to do work related things.
Lot of effort to go through to do personal stuff at work, uhm, when you should be working.
Point well taken. However, I'm on call 24/7, and it just would make it easier if I didn't have to say "I'm doing work stuff, so connect to that VPN. Now I'm doing personal stuff, so connect to my personal VPN provider." Also, sometimes I have to be online and connected to work, but really only to listen in and jump in if an issue arises that I'm responsible for.