Varnishing existing floorboards
You might try a glaze coat and then over-seal with more varnish.
Some background:
A pigmented stain is made from 'bits' of color (the pigment).  When applied to bare wood, most  of the color is on the surface.  (But HerrBag, I hear you saying, my Minwax stain is oil based and penetrates.  Alas, Minwax is a MIX of dye and pigmented stain.
Most paint is pigment mixed into a base material that lets you disperse it evenly.  The opaqueness comes from the concentration of pigment and the base material itself.
GLAZES
are the same pigment in a transparent base that lets you 'float' a color layer over previously finished wood.  Indeed, to make a custom glaze, you add a small percentage of regular paint to the glazing base.
To adapt the technique for large areas will require the slower setting version glaze and a swivel head applicator (as used by flooring pros, its an oversized tubular paint pad (like a paint roller that doesn't rotate)).

A good article on glazes and glazing from Woodworkers Journal is here.