The deal was agreed upon by both teams a hour before the trade was made official as Everett had to approve the deal.
Along with Everett, the Rangers have agreed to pay for the remainder, or at least most of, Everett's salary, which is close to $5 million.
The White Sox gave up three minor leaguers, which will be chosen by the Rangers from a pool of eight players. It us unknown if lefty Neal Cotts is in the pool.
Everett, 32, is batting .274 with 18 homers and 51 RBIs in 74 games this season. He has split time between all outfield positions, but it looks as though most of his playing time with the Sox will come in center field.
Everett is expected to be in the starting lineup, along with Alomar, on Wednesday against the Twins.
With the trades, the White Sox have designated both Armando Rios and D'Angelo Jimenez for assignment, which likely result in their departure from the organization.