It looks like custom fields are getting stored as a config within the Db. Is there a way to track these under git and have them deployed through Jenkins? Are there any 3rd party libraries that can help with this? Such as something like Unicorn Sync or TDS for Sitecore?
The way to do that is through the export of the structure from the Administration menu in the backend.
You will be able to export a file for deployment that contains all the changes that occurred in the structure including "Custom fields" to existing modules and you will need to place that file under App_Data/Deployment in the new site to load the changes.
That file of course can be tracked in GIT and you can set CI on as well with pre and post build actions.
Hope that helps