#!/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 matching
for my $day_of_year (@day_of_year) { 
  print "looking for things in $day_of_year folder /home/myfiles/day_of_year/$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/*/desired_file.txt
/home/myfiles/day_of_year/4257/a/desired_file.txt                is the file name


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


found a file!!!!!!
lines:
data
/home/myfiles/day_of_year/4257/c/desired_file.txt                is the 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/*/desired_file.txt
/home/myfiles/day_of_year/4257/a/desired_file.txt                is the file name


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


found a file!!!!!!
lines:
data
/home/myfiles/day_of_year/4257/c/desired_file.txt                is the 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/*/desired_file.txt
/home/myfiles/day_of_year/4258/a/desired_file.txt                is the file name


found a file!!!!!!
lines:
data
/home/myfiles/day_of_year/4258/c/desired_file.txt                is the 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/*/desired_file.txt
/home/myfiles/day_of_year/4259/c/desired_file.txt                is the file name


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