I would personally suggest to update to Spine 3.7, both editor and runtime side. The runtime API and file formats have not changed a lot, so the process should be mostly painless. Please note that we Spine 3.7 is still in beta, however, the release is scheduled for early January. You can find the 3.7-beta Spine Runtimes here GitHub - EsotericSoftware/spine-runtimes at 3.7-beta. The corresponding Unity package can be found here Spine Unity Download

For the runtime, I would go for 3.7 as soon as you're able. Lots of fixes and better implementations of things.
See the upgrade guide from 3.6 to 3.7 specific to Spine-Unity for more information on what's changed: Spine-Unity 3.6 to 3.7 Upgrade Guide
We recommend you follow the steps there to take advantage of the new folder structure that allows you to use Unity's new Assembly Definitions feature for Unity 2017.3 and up.

Note that if your old skeletons were exported as jsons with 3.6, they will still be readable with the 3.7 runtime so there's no rush having to re-export all your skeletons in that case. Though now may be a good time to set up a batch file to export your skeletons if you haven't already.