Laird will remain the No. 1 starter, but Daniels said Melhuse would play about two days a week to give Laird a break.
Daniels said the team would send backup catcher Chris Stewart to Triple-A Oklahoma so he has an opportunity to play everyday.
"[Chris] needs to play more," he said. "Where he is in his development, we think this will help him."
Melhuse, who has been with the A's for five years, thinks Washington had some role in getting him to Texas.
"I'll do whatever Wash wants me to do -- he's always been real good to me. I'm anxious to get over there and play. ... I'm 35, so I am not getting any younger, but I think I still have some things to offer.
"I'm excited about the opportunity. These things are part of the game. Hopefully, this will be an opportunity to play."
The Rangers' catchers have struggled this season. Laird's pitch-calling ability has been questioned. Stewart has played in 16 games, with 11 starts.
Both have struggled offensively as well. Before Saturday, Laird was hitting .233 with three home runs and 20 RBIs. Stewart was a .243 hitter with three RBIs.
Melhuse was used seldom this year in Oakland, appearing in 12 games with 26 at-bats.
"I wasn't asking to be a starting catcher, just a normal Major League backup who plays every five days or in a day game after a night game," he said. "Jason [Kendall] is a great player, and he's struggling a little bit this year, but it was evident [even] with his struggles there wasn't any room here for me."
Drew Davison is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.