Although there is (currently) no option within New Outlook for this, as an alternative you could consider using the Files companion app from Microsoft;
Get started with Microsoft 365 companions
This app keeps track of recent activities for files you have access to and also allows you to quickly copy/share a link or open its file location so you can attach it more quickly within New Outlook.
It's not a 100% replacement but might give some relief over the native implementation of New Outlook. Note that this is for M365 for Business only.