I haven't found a definitive 'no' but with a bit of quick googling - the current mission extension runs until (at least) 2021 and there is very little onboard fuel for significant course updates. Some of the other potential targets were in fact ruled out by the selection of Ultima Thule. The extended mission for New Horizons calls for the spacecraft to conduct observations of, and look for ring systems around, between 25 and 35 different KBOs.

I'm curious what is the reason to think there would be ring systems around these objects? Aren't rings usually associated with planets much larger than earth, whereas these objects are much smaller?

Long-lasting rings would be - fundamentally all you need for a ring to form is some debris and a gravity well.

To be clear, something that "doesn't last long" on an astronomical timescale could still quite happily exist stably for several dozen human lifetimes.

They have to find another object first. They'll be looking using LORRI, but it was hard enough finding MU69 that the probability is pretty low. Regardless, they'll also be using NH like we still use the Voyagers, for instrument readings on the outer solar system as long as we can still receive signals (and the spacecraft is still operating).

Nothing concrete, but there probably will be another https://www.space.com/42808-nasa-new-horizons-possible-third...

Your reference says that there will be continued exploration, but doesn't mention another actual fly-by. Funding beyond 2021 isn't confirmed, and even if they top up the funding, they can't top up the manoeuvring fuel.

Well, in the other poster's defence, I thought "nothing concrete" covered things well enough, but these quotes from the article seem sanguine about the prospect:

"And we hope to hunt down one more KBO — one more Kuiper Belt object — and make an even more distant flyby in the 2020s," Stern added [...] New Horizons [...] has enough juice [...] to keep the power on through the mid to late 2030s, he said.”

The long range camera may help them see another.

Lets hope.