It's not if, it's eq; the string "abc" is not equal to " abc" (computers are quite pedantic that way). If stripping white space is causing problems elsewhere then create a stripped copy and compare that instead.

( my $staff_last_clean = $staff[ $i * 3 + $staff_last ] ) =~ s/^\s+//;
if( $staff_last_clean eq $names[ 0 ] ) {
  # ...
}
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That'll leave the original untouched for elsewhere.