Why don't you post some detail photos of the setup, so we can all get on the same page.
Note, newbies have 5 posts per day. To attach photos, go to the "Advanced Editor", and select the paper clip Attachments.
You may well have some issues. A 7 speed can either be a freewheel, or a cassette, but if it is a cassette, an 8/9/10, or an 11 speed cassette may not fit on the freehub. So, hopefully the sprocket cluster is part of a wheel that could be rebuilt.
There are some new 9, 10, and maybe 11s freewheels. I think DNP Epoch makes a few. But, whether they'll fit will still depend on your configuration.
What size of wheels are you using? Another issue may be ground clearance if you get a huge cassette, and a MTB derailleur. Or, is the derailleur mounted above the wheel? That may make it easier.
You could certainly get as good of, or even better range of gearing if you go with say a 11/46 cassette, and choose an appropriate chainring. One issue you do run into with the huge cassettes is that they tend to have large gaps between sprockets... so, say 11/13/15/18... instead of going 11/12/13/14... It may or may not make a difference for your riding, but a handcycle may benefit from easy jumps between gears.
You could also choose a large cassette, and keep a double, and maybe triple chainring configuration to get more gearing, but one quickly runs into a limitation of rear derailleur capacity. Bailout gears?