I don't have the time to delve into the depths of Google Books, Internet Archive nor search through the archives of American newspapers, but a cursory view suggests that the war in Vietnam provoked this surge in use
In February 1968, after the Vietcong attack on Saigon and after the North Korean seizure of the Pueblo, Gallup reported that the hawkish rating rose from 56 to 61 percent in one month. A "dove" stance would have been particularly unpopular with the college-trained who, according to Gallup, "have been found to be more hawkish than those with less formal education." (N.Y. Times, February 17, 1968.) To campaign for President as a "dove," Kennedy would have to fly against the wind.
The American Federationist
… hearings had increased the hawkish element in our country, and the hawkish pressures, I believe he said something to that effect, I read in the paper this morning. Dr. Morgenthau. Yes ; I read this, too. The Chairman. How do you interpret that ? Dr. Morgenthau. Well, you put me on the spot. [Laughter.] Not everything that emanates from the White House is equivalent in truth to what has emanated from Mount Sinai.
U.S. Policy with Respect to Mainland China
One would expect a majority of Japanese to opt for a moderate position neither hawkish or dove-like, but again we find that most of those with any opinion on the Vietnam conflict disagree with the Sato policy. Twice as many Japanese advocated immediate United States withdrawal as wanted limited defense of South Vietnam or total victory over the north, and many 1966 subgroups were even more opposed to the Tokyo support for Washington policy.
Asian Survey - Volume 7 - Page 455
Talks with hundreds of citizens from all corners of the country, all walks of life, show clearly that hawks are becoming more hawkish, doves more dovish. The great mass of Americans in the middle-—those who are tormented by the war but honestly don't have any idea of how it can be ended with honor-— is slowly shrinking as many of those once in its ranks begin to choose sides.
Congressional Record (1967)
The adjectival form, hawkish, is derived from the expression “doves and hawks“ that arose during the Cuban missile crisis in 1962
Like his brother, the president, RFK originally favored an air strike against Cuba. But they both quickly changed their minds and came down on the side of the blockade option outlined by McNamara.
The term “doves and hawks” can be traced back to the missile crisis.
When it was all over, the leader of the hawks, Mac Bundy, had the grace to concede that the doves had got the better of the argument.
“Everybody knows who were the hawks and who were the doves,” Bundy told the ExComm on the morning of October 28, after Khrushchev announced that he was withdrawing his missiles. “Today was the day of the doves.”
Foreign Policy.com Article posted online OCTOBER 11, 2012,