I found a number of Cook's listed in the book "The Ship's Locator" - so far I haven't found any first names listed with the Cook surname - I will list below what I found and you can try to see if any of these are "your Cook."
ooops - the first Cook I found was at sea before your Cook was born!
2. A schooner names Ida and Annie had a Cook who was a Master in 1875 - this ship arrived in New Hampshire, USA on July 21, 1875
3. A schooner Andrew Peters had a Master named Cook - also arrived in US in 1875
4. In 1880 the ship Jed F. Duren had a Cook for a Master
5. Schooner C.C. Clark with Master named Cook who was from Baltimore wrecked in the Caribbean Sea
6. A Machias Cook on the schooner Willie Putnam in 1880
7. The Bark St. James had a Cook for a Master
8. The schooner Sarah Ann Carpenter had a Cook for a Master in 1898
There probably were hundreds of Cooks in crew lists but none listed in the few books I have on hand.
Have you tried the following websites?
www.maritimeheritage.org/log.html
www.www.eicships.info/ships/shipsearch.asp
www.usmm.org/links.html
www.eicships.info/seafarers/index/html
Good luck