Is there any way to stain white pine to match a natural red pine?
I have had good success making pine look like maple, walnut, teak or semi-transparent blends by mixing deep colors and adding latex paint to an oil based stain to make it more pastel like to tone down the contrast. I even made K3 particle board stain like old mahogany with many thin layers of brush grain stain and urethane for a dining room table extension . Usually I might brush on and wipe off excess for contrast  or apply a sealer first lightly for less contrast.

If you like my idea towards a slightly semi-transparent stain rather than a clear stain consider taking 1/4 cup in  yogurt container then add teaspoon of latex white or any colour you are inclined towards . Then experiment on a hidden side with untinted and tinted then add more latex stir and try again until you get a blend of clear stain and semi-opaque or "milk-stain". 

THe above are the typical mini-wax stains.