Chain tension and tensioning a chain

I ride a SS and the chain slipped off earlier this evening. I managed to sort it out but I wanted to get some advice. Tightening the nuts takes strength which I don't seem to have whilst using a bog-standard spanner. The rear wheel still has a little movement despite my best efforts , whilst the front wheel is fixed firmly. Normally I'd take the bike to my, very helpful, lbs and get them to tighten the nuts but the last time that happened I had to get them to loosen the nuts so I could fix a punture. Clearly that isn't ideal.

I'm right-handed and have arthritis in that shoulder so applying a lot of pressure isn't a good idea any more. Would a socket wrench be better for me? Is there anything else I could use that gives great results with less effort?

TIA

Are your cones tight enough? is the play 'side to side'? a little adjustment to the bearings might be necessary.