We present a general method to modify existing uniquely decodable codes in the $T$-user binary adder channel. If at least one of the original constituent codes does not have average weight exactly half of the dimension, then our method produces a new set of constituent codes in a higher dimension, with a strictly higher rate. Using our method we improve the highest known rate for the $T$-user binary adder channel for all $T \geq 2$. This information theory problem is equivalent to co-Sidon problems initiated by Lindstr{ö}m in the 1960s, and also the multi-set union-free problem. Our results improve the known lower bounds in these settings as well.