Here's some evidence to that:
"A Night To Remember" by Walter Lord (pg. 89)
(Captain Smith has just told Phillips and Bride to forget about SOS and try and save themselves)
"Captain Smith returned to the boat Deck, walked about speaking informally to men here and there. To Fireman James McGann, 'Well, boys, it's every man for himself.' Again to Oiler Alfred White, 'Well, boys, I guess it's every man for himself.' To Steward Edward Brown, 'Well, boys, do your best for the women and children, and look out for yourselves.' To the men on the roof of the officers' quarters, 'You've done your duty, boys. Now, every man for himself.' Then he walked back on the bridge."
Walter Lord interviewed many survivors for his book; so I would say some credit should be given to his book.
I double checked on this website, all of the individual men Capt. Smith apparently briefly spoke to survived.
To my thinking, Smith probably did not call out into a bullhorn the last order a skipper can give, but it would seem in character of him to announce it to just small amounts of his crew.