In my recent tests, I've starting playing around with changing the actual style prompt as well as using metatags in the lyrics prompt, and it seems to take this into account. For instance, say I'm making an Alternative song, the style prompt might read: "Alternative, Alt-Rock, Male Vocal" after a verse and chorus, I want a jazzy bridge, so in the extension where I have the "[Bridge]" metatag in the lyrics prompt, I also erase and change the style prompt to read: "Jazz, Male Vocal" or sometimes "Jazz Bridge, Male Vocal" I've had decent success with this, but like with all things, it's still a little iffy.
If you're light on tokens, it's worth noting that sometimes it will take an additional extension with the new style to give the AI time to shift gears, in my experience. So it may be better to change the style prompt and leave room before the actual bridge you want to jazzify, for instance with an [Instrumental Interlude] or something along those lines. This also applies to rhythm changes, adding instruments, or changing mood/key/mode.
As always though, all of these tips should be taken with the caveat that sometimes it just will not do what you want. It's still early in it's lifespan, and it will probably never be perfect.