I'm not familiar with Winchester bolt faces, but from what I've seen on other rifles, there is a plunger ejector in the face. You might try cleaning it thorougly as any gunk down in there can cause it to work marginally. (also makes the firing pin work better too!).

Other than that, it could be a faulty ejector spring, or the extractor may be too tight on the rim of the case. I've seen that happen where the rotating of the bolt will cause the ejector to dig into the soft brass material and then hang on for dear life. It may just need to be cleaned, or relieved a little.

Those things are my best guess.