This is, of course, a matter of personal opinion to a large degree. My own view is that the Southern Baptists continued to grow after 1962 as a reaction to the growing liberalism of American life and American culture.
The Southern Baptists are known mainly for their very conservative social beliefs and their staunch fundamentalism. Both of these views seem to be in some ways inconsistent with modern American culture. However, I would argue that this inconsistency is exactly why the Southern Baptists have increased in number. Their success is part of a backlash against modern America.
In the last 50 years, America has become more liberal on a broad variety of social issues. One such issue is the issue of race. America has become inclusive and tolerant to such a degree that some whites feel that they are under attack in our country today. Another issue is that of the role of women in society. We have moved far away from the time when male dominance over women was accepted as a matter of course by most Americans. Finally, there is the issue of homosexuality. Change on this issue has happened much more recently than on the other two issues mentioned here, but anti-gay ideas are definitely on the decline in modern America.
All of these changes are threatening to traditional Americans. Americans who hold strongly to traditional values often feel that the country is being taken away from them. As a result, they want to fight back in an attempt to maintain what they see as proper moral and social values. I would argue that this is why the Southern Baptist church has grown. This church offers moral certainty and support for those who feel that American culture has gone in the wrong direction. As such Americans feel more and more threatened, they are more likely to move to churches that offer them the feeling that their values are still accepted and welcomed.