The standard way to do what you want is to open and read one file at a time, keeping whatever data you care about in variables. The example code you gave (logically) does this - it reads every line from FILE1, then FILE2, then FILE3. This is because || is short-circuited: as long as the first condition is true, the whole expression evaluates true. This keeps happening until the end of the first file, etc. (it's safer to check line existence with defined though).

Instead you could read in from standard input with a simple while ( <> ) { ... } and pipe input to the program from elsewhere. Or you could take a list of filenames as arguments in @ARGV and process them individually:

for my $file ( @ARGV ) {
   open my $fh, '<', $file or die "$file: $!";
   while ( <$fh> ) {
      ...
   }
   close $fh;
}
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Perhaps I should ask: is there any particular reason you need all filehandles open at once?