Or don't? Because the other company may hire you at the higher level anyways, that's the point. Because you're getting multiple companies to compete for the experience you already have. It's your task to demonstrate in interviews that you have the skills for that higher level. Which is a huge shortcut.

And "maybe your current company will surprise you" just sounds like being taken advantage of to me. Because the reality is they probably won't. Not at anywhere near the same speed, usually.