"Is there a way to be sure to receive the updates for this version of the libre-kernel?"

I assume that you had to download the package manually and install it. To be sure you get updates (now that your ethernet somewhat works, even if slowly) you should now go back to https://jxself.org/linux-libre/ and set up the repository properly, following all the directions there... including installing one of those packages listed in the chart at the end. Because it's those meta packages at the end that important. But your computer also needs to have the repo properly set up, and one of those meta packages installed from it. So do everything shown there from start to finish, as if you're just starting out.

As an example: Your message shows you were using 4.8.12. well, 4.8.13 is now out. Once you've done all of those things, getting to 4.8.13 should be as simple as:
sudo apt update
sudo apt full-upgrade
(And then rebooting once everything is done.)

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