Ha, I was in a pretty similar situation myself.

This was my first time asking someone out. I handled the initial rejection fine, but I made the mistake of trying to "right the wrong" and said some kind of cringy stuff trying to force a friendship afterward. It made the situation very awkward! It's taken some time but I've finally accepted the fact that 1) I did nothing wrong and 2) it isn't/wasn't up to me to fix anything. I wish I had realized that sooner, but experience help you grow I guess.

Anyway, you can always flirt with her to see if she's receptive. Imo you're only going to regret it if you don't try, but you're going to have to be willing to put that friendship on the line for something better. It might be a little uncomfortable for some time but if you keep acting the same way towards her as you did before you asked it won't be awkward unless one of you makes it that way (basically just read what I did above and DON'T DO THAT LOL).

Like other people said though, since it's a work thing you can probably take it a bit slower. You're going to see her a lot.