I realize that an answer was already accepted for this, but from a "strictly *nix purist angle" it seems like the right tool for the job is pcregrep, which doesn't seem to have been mentioned yet. Try changing the lines:
echo $f | grep -oEi '[0-9]+_([a-z]+)_[0-9a-z]*'
name=$?
to the following:
name=$(echo $f | pcregrep -o1 -Ei '[0-9]+_([a-z]+)_[0-9a-z]*')
to get only the contents of the capturing group 1.
The pcregrep tool utilizes all of the same syntax you've already used with grep, but implements the functionality that you need.
The parameter -o works just like the grep version if it is bare, but it also accepts a numeric parameter in pcregrep, which indicates which capturing group you want to show.
With this solution there is a bare minimum of change required in the script. You simply replace one modular utility with another and tweak the parameters.
Interesting Note: You can use multiple -o arguments to return multiple capture groups in the order in which they appear on the line.