After reading about it online, I gave a version of layering clematis a try.

The version I read about involved running a clematis vine through a pot of soil, instead of just burying part of a vine in the soil.

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You take a newly sprouted vine (so it is still flexible enough to bend without breaking) and bend it into a "U" shape and bury it in a pot of soil. Over the course of the year, the vine will take root in the pot and the following spring you will see new growth coming out of the pot.

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Partially burying the pot in the soil keeps the forming roots cooler, which clematis roots like. Covering the pot with leaves and pine straw helps reduce the likelihood that the squirrels will dig the vines out of the pots and helps keep those roots cool.

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The results have been very successful provided the vine remains in the pot, which my squirrels seem determined to prevent. I think I spent more time and effort reburying vines than had been dug out of the pots than anything else.

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Here is a link that might be useful: Propagating Clematis