Savage made bolt-action .30-30 repeaters, that "might" be a little more accurate than a break-open SS - the old Model 340, newer Model 40, IIRC.
IIRC, Rossi also makes/imports a .30-30 break-open SS rifle.
Of couse, for an accurate SS .30-30 rifle, a gennie Winchester Model 1885 High Wall (or a newer clone), or a Charles Daly Little Sharps might be located.
They're out of your stated price range, though.
.