Regular expression for odd binary numbers without leading zeros
First you might just add that single $1$ to your expression as an alternative to what you already have. One obtains $1 \cup 1(0\cup 1)^*1$.

An alternative is $1(0^*1)^*$. That one is shorter, but not necessarily better. 

Actually that expression I obtained by starting with the FSA below, and realizing that the rightmost two states could be replaced by iterating $0^*1$.