If you have all of these files in one folder and you're on Linux you can use:
rename 's/test-this/REPLACESTRING/g' *
The result will be:
REPLACESTRING.ext
REPLACESTRING.volume001+02.ext
REPLACESTRING.volume002+04.ext
...
rename can take a command as the first argument. The command here consists of four parts:
s: flag to substitute a string with another string,
test-this: the string you want to replace,
REPLACESTRING: the string you want to replace the search string with, and
g: a flag indicating that all matches of the search string shall be replaced, i.e. if the filename is test-this-abc-test-this.ext the result will be REPLACESTRING-abc-REPLACESTRING.ext.
Refer to man sed for a detailed description of the flags.