Our plugin strores its own data in separate database file, so by making a backup using OOTB Jira functionality doesn't really backup TheScheduler data.
Starting from TheScheduler 1.6 (please download from Atlassian Marketplace) there is additional action which allow user to make a backup of all existing scheduled issues. This backup can be restored then by using restore functionality.
I don't know if I understand you correctly, but if you mind that you have two the same Jira instances in terms of installed version of our plugin (there was for example TheScheduler 2.0 in the backup version of your Jira, and currently you are using still the same 2.0 version) then you could copy the following directory "<JIRA_HOME>/data/plugins/" from backup to current instance.
Please let me know If I correctly addressed your problem.
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