It is not what the OP asked for.

He wants to replace the second Fate| substring, whereas your solution replaces the rightmost Fate| substring.

If there are only two Fate| substrings in the string, it is equivalent; but if there are more than two, then your solution is wrong.

Thus your solution is actually not more general, but rather more restrictive since it pre-supposes that there will only be two Fate substrings in the string!

But do not be disappointed, we are all here to learn and using index was a good idea. Have a look at the following small change to your program:

my $match_pos = index $str, $search, (index $str, $search) + length $s
+earch;
[download]

See what it does? I starts looking for $search after the first occurrence of $search.