Well, you have to get rid of the weeds for any groundcovers to establish well. Start with a small area, clear it and plant something like Geranium macrorrhizum--Big Root Geranium. Plant several starts about 2 feet apart. They'll fill in. Clear around them, take starts and plant those. You'll be amazed how fast it goes.

Big Root geranium tolerates shade, drought, tree roots. It's highly weed suppressing once established, deer resistant, smells good and is very easy to control.

You can suffocate the weeds by covering them with 6 layers of newspaper, then a 1" layer of topsoil, followed by a 1" layer of compost. Seal the edges of the newspaper by covering them with soil, bricks, etc. I've found that ceratostigma plumbaginoides (plumbago) works well for situations like the one you describe.

The leadwort mentioned is indeed a handsome groundcover, but it dies back to the ground in winter and the old growth must be mowed or cut back every year.

Pictures? One from the front looking towards the rear, and the reverse.
Do any significant rooms look out onto the side yard?
Is there through traffic? Would you like to have a quiet nook?

And how big is this area? Width x length

Look up "narrow yard" on Pinterest or Houzz and you will see a ton of ideas from simple to elaborate. they are a challenge but can be extremely satisfying.