The mountains become lighter, and float higher.
In our world, the Appalachians formed from plate tectonics in the remote Ordovician. But much later, in the Cretaceous, they rose again. Plate tectonics were not responsible for this late rise. Instead, the mountains rose because they became lighter.
Appalachians Get a Face-Lift From Earth's Mantle
Shrinking due to cooling.
As the deeper layers of a planet cool down, the planet shrinks. Of course this will result in stresses on the surface, which has to adapt to a lower volume.
This will produce wrinkles on that very surface, and those wrinkles are the renewed mountains you talk about.