Furniture, Wood and Cabinetry Finishing - Refinish Handrail

I have stained and varnished hardwood handrails for the stairs in my house. After many years, the varnish AND WALNUT-COLOR STAIN have worn off along the sides where the rails are grabbed, and slid over, with hands. Unfortunately, I didn't notice this until the damage was done, and the light wood underneath now shows through in 'steaks'. I have tried to lightly sand the areas, and cleaned them with mineral spirits, but the stain won't take. I'm guessing that the body oils have sealed the surface pretty well!

How can I re-stain those areas without having to remove the varnish, resand, and restain the whole handrail? I understand that I will have to revarnish the whole thing - that's OK!

Thanks for suggestions.