There are seats back on the 21st and 22nd of June, but I can't see anything going out unless you head over on the 14th where CO19 and 29 both have space.

The continental website lets you search for award seats without logging in. There's always the scope for some kind of blocking for the airline's own FFs but you're far better off ringing up and saying "I want seats on CO29 on the 14th" rather than "have you got anything from New York to London week commencing June 20th"