In smaller companies where decision makers are usually more empathetic towards the employee, there are various other ways you can get them to move without having to use an external offer as leverage. At bigger companies that are more restricted by process, they do this for a reason and ultimately data speaks louder than words. Assuming you've exhausted the usual strategy of establishing your value that is not fully compensated and they're just not moving, if you have a competing offer for $20k more, they now have no choice but to increase the urgency with how they approach you up the chain.

This is very important. You have to make sure you have the upper hand. This has been successful for me even in the private sector. I left a department once with another offer in hand, and they were willing to rewrite the job spec (and salary range) to convince me to stay. The other offer was more lucrative with more upward mobility, so I ended up leaving anyway, but it's a much better experience when you don't have anything to lose.