Honestly, you have a number of things to work with. Professional baseball teams popped up all over the place and not just what would be considered major league. The American and Federal Leagues were the only major leagues founded in the 20th centuries. Perhaps the most curious aspect of the Federal League's collapse was that unlike with the American Association of the 19th century, the remaining major league (or in this case, leagues), did not absorb teams from the former major league.
The St. Louis Cardinals, Los Angeles Dodgers, Cincinnati Reds and Pittsburgh Pirates are all relics of the former American Association. Perhaps the American and National Leagues could have absorbed some of the teams from the Federal League to create new histories for major league teams? Since some of these teams were already in major league markets, it's possible that a few old franchises could have been revived.
Baltimore Terrapins: It's not as if Baltimore was an alien baseball market. Long before the Browns moved there, Baltimore lost the AA/NL Orioles to contraction in the National League and the original charter AL franchise that would move to New York and become the Yankees (irony considering Ruth was from Baltimore). Baltimore was thus ripe ground for the Federal League, but the Terrapins could easily have been absorbed by the American League to open a potential rivalry between the Washington Senators and the Baltimore Terrapins.
Chicago Whales: Given the situation in New York before the 1958 season, it's not inconceivable that you could have seen another major city with three major league teams. They already have their own ballpark, which would later become known as Wrigley Field. Now imagine the Whales in the National League, creating a similar rivalry that the Giants and Dodgers had when they played on the east coast, only this is between the Whales and Cubs.
Kansas City Packers: They were locked in a pennant race with the Whales towards the end of the final season of the Federal League. Given NL saturation in the area (Cubs, Cardinals and now the Whales), the AL could probably get this franchise. There's no guarantee that the Packers would remain in Kansas City, but this franchise is ripe for the plucking.
Buffalo Blues: Buffalo might seem obvious. They had a Federal League team and I think the NL could probably do well with this franchise. An astute businessman could remain this team the Bisons to harken back to the old major league team. Unfortunately, I don't think Buffalo would be able to keep supporting a major league team and don't be surprised if they move out west.
So let's review. For the 1916 season, here's how the spread would look like post FL:
National League
Boston Braves
Brooklyn Robins (not known as the Dodgers in 1916)
Buffalo Bisons (fka Blues)
Chicago Cubs
Chicago Whales
Cincinnati Reds
New York Giants
Philadelphia Phillies
Pittsburgh Pirates
St. Louis Cardinals
American League
Baltimore Orioles (fka Terrapins)
Boston Red Sox
Chicago White Sox
Cleveland Indians
Detroit Tigers
Kansas City Packers
New York Yankees
Philadelphia Athletics
St. Louis Browns
Washington Senators
How about more teams in the south, maybe a team in New Orleans, Mobile, Little Rock, or Chattanooga. The Chattanooga Choo Choos?
FWIW, New Orleans had the chance at Major League Baseball twice. They could have gotten the Athletics from Oakland in the late 1970's or the Pirates from Pittsburgh in the early 1980's. New Orleans actually has a pretty interesting history with professional baseball as it was home to the minor league Pelicans until the late 1950's (again in the 1970's) before the Zephyrs moved to New Orleans from Denver in the 1990's. Shoeless Joe Jackson played for New Orleans during one of their championship seasons. New Orleans has obviously missed its window because no one in their right mind would want to put a team there now, but back in the late 1970's and early 1980's, the Louisiana Superdome looked very attractive. Now, another disaster like Katrina could seriously disrupt operations unless someone is willing to front funds to upgrade Zephyr Field in the suburb of Metarie.
As far as Mobile, Little Rock and Chattanooga go, they all have professional teams. Mobile has the BayBears (AA affiliate of the Diamondbacks), Little Rock has the Travelers (AA affiliate of the Angels) and Chattanooga has the Lookouts (AA affiliate of the Dodgers). New Orleans also has the Zephyrs, the AAA affiliate of the Marlins.