A few other suggestions in addition to what Rob advised on this:
1.) We recently introduced the SAP Jam activity hub, a desktop client that shows aggregated feed events from SAP Jam and IBM Connections. You could use that as a mechanism to slowly migrate from Connections to Jam, and provide less of a jarring experience in moving between the two for end users.
2.) We also recently introduced the SAP Jam file sync client for Microsoft Windows. This would allow you to synchronize large folder/file structures from other repositories.
3.) You could also consider using the "extract zip" file option, which allows you to unzip a single file in Jam and retain the file folder structure/hierarchy.