Is it possible to add a second switch to a light, without unearthing the wires to the existing switch
There are smart switches that can do this. The issue will be you have to purchase both quite expensive and they take full sized openings (larger box will be needed at the stairwell single gang box) I cannot remember if they made one without a neutral required but most balk at the price of ~70$ just for the electronic 3/4 way switch. master snd slave required or controller and remote for those offended by the first term. I believe the series is caseta  by lutron the current price 60$  Theee were others that did not require hubs same MFG that I have used.
You need to verify if a neutral is available or needed by the switch some require neutral some require ground to operate.