Despite your updates, you still haven't shown us the code you are really running so I'm still not entirely sure what you are trying to accomplish but this should give you a push in the right direction.

#!/bin/env perl
use strict;
use warnings;

my @day_of_year = (4257,4257,4258,4259);

# If @day_of_year only contains the days then there is no reason for m
+atching
for my $day_of_year (@day_of_year) { 
  print "looking for things in $day_of_year folder /home/myfiles/day_o
+f_year/$day_of_year/*/desired_file.txt\n";

  # there may be multiple directories containing desired_file.txt
  my @desired_paths = glob('/home/myfiles/day_of_year/'.$day_of_year.'
+/*/desired_file.txt'); 

  for my $info_desired_filepath (@desired_paths) {
    print "$info_desired_filepath \t\t is the file name \n\n";
    <STDIN>;
    open (my $info_FH, '<', $info_desired_filepath)
      or die "cannot open $info_desired_filepath..........\n\n";
    print "found a file!!!!!!\n";
    chomp(my @lines = <$info_FH>);
    close($info_FH);
    print "lines:\n";
    print "$_\n" for @lines; # yep, we actually read the data
  }
}

__DATA__

Directory structure:
  myfiles/
    day_of_year/
    |-- 4257
    |   |-- a
    |   |   `-- desired_file.txt
    |   |-- b
    |   |   `-- desired_file.txt
    |   `-- c
    |       `-- desired_file.txt
    |-- 4258
    |   |-- a
    |   |   `-- desired_file.txt
    |   `-- c
    |       `-- desired_file.txt
    `-- 4259
        `-- c
                `-- desired_file.txt

Output:

skimming for the day of year number: 4257
looking for things in 4257 folder /home/myfiles/day_of_year/4257/*/des
+ired_file.txt
/home/myfiles/day_of_year/4257/a/desired_file.txt                is th
+e file name


found a file!!!!!!
lines:
data
/home/myfiles/day_of_year/4257/b/desired_file.txt                is th
+e file name


found a file!!!!!!
lines:
data
/home/myfiles/day_of_year/4257/c/desired_file.txt                is th
+e file name


found a file!!!!!!
lines:
data
bogus
data

skimming for the day of year number: 4257
looking for things in 4257 folder /home/myfiles/day_of_year/4257/*/des
+ired_file.txt
/home/myfiles/day_of_year/4257/a/desired_file.txt                is th
+e file name


found a file!!!!!!
lines:
data
/home/myfiles/day_of_year/4257/b/desired_file.txt                is th
+e file name


found a file!!!!!!
lines:
data
/home/myfiles/day_of_year/4257/c/desired_file.txt                is th
+e file name


found a file!!!!!!
lines:
data
bogus
data

skimming for the day of year number: 4258
looking for things in 4258 folder /home/myfiles/day_of_year/4258/*/des
+ired_file.txt
/home/myfiles/day_of_year/4258/a/desired_file.txt                is th
+e file name


found a file!!!!!!
lines:
data
/home/myfiles/day_of_year/4258/c/desired_file.txt                is th
+e file name


found a file!!!!!!
lines:
data
bogus
data

skimming for the day of year number: 4259
looking for things in 4259 folder /home/myfiles/day_of_year/4259/*/des
+ired_file.txt
/home/myfiles/day_of_year/4259/c/desired_file.txt                is th
+e file name


found a file!!!!!!
lines:
data
[download]

Updated: Added directory structure and output from my test run.

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