"woolen hat" is a correct and usable part of a sentence in written English.
You can use it to describe a type of hat, e.g. "I bought a new woolen hat to wear during the cold winter months."
I still don the donkey jacket and the black woolen hat now and again.
Mr. Mohammad, wearing a woolen hat in the midday heat, stared into the distance, smiling.
Mr. Muhammad was wearing a woolen hat, as was Nawid, just donated by a Japanese woman.
He walked to the front of the store, shaking his head, which sported a woolen hat dusted with plaster.
One morning last spring, Mr. Ryan stopped Josh Brignoni, who had failed to remove his blue woolen hat.
Except for a woolen hat and a bloody bone, there will be no clues to her disappearance.
Who are they, with their parkas and scarves and woolen hats, their boots and sneakers and binoculars and notepads?
He gazes at 1,200 people in overcoats and woolen hats and snow boots and asks for a show of hands.
And, where practicable, the young adults shall wear woolen clothing, and woolen hats, and they shall smoke unfiltered cigarettes, and their faces, while unwashed, shall shine with dignity.
At each, the family matriarchs were sitting outside in their Tibetan garb, knitting the woolen hats that are so popular among the trekkers who come here.
Almost in unison, the inmates, their heads covered with shawls or woolen hats against the chill, reported that everything was "normal".
