How to add Multiple Searches in AWK command

You can match on multiple patterns like so:

awk '/Jun/ || /July/ || /Aug/' <(ls -lrh)

This returns all matches for either Jun, July, or Aug. You don't require the print statement as that is awk's default action.

As a general rule, it is a very bad idea to parse the output of ls. A better way to do what you want is

stat -c "%n %y" * | grep 2013-0[678]

Alternatively, check only the last field to protect against the unlikely case where the file name itself is a date:

stat -c "%n %y" *| awk '$NF ~ /2013-0[678]/' 

From man stat:

   -c  --format=FORMAT
          use the specified FORMAT instead of the default; output  a  new‐
          line after each use of FORMAT

   %y     time of last modification, human-readable
   %n     file name

ls is a command to display information about files in a human readable. Do not use it in scripts.

The command to find files in a directory tree matching certain criteria is find. If you have GNU find (i.e. non-embedded Linux or Cygwin) or FreeBSD find (i.e. FreeBSD or OSX), you can use -newermt to match files that were last modified after a certain date.

find . -newermt '1 June' ! -newermt '1 September' -print

Add a year if you want to match files in a specific year, rather than the calendar year when the script runs.

If your find implementation doesn't have -newermt, you'll have to make do with -newer, which compares the date of the file that find found with a fixed file. use touch to create temporary files with the date that you want to use as a boundary.

dir=$(mktemp -d)
touch -d 201306010000 "$dir/from"
touch -d 201309010000 "$dir/to"
find . -newer "$dir/from" ! -newer "$dir/to" -print
rm -r "$dir"