I'd advise you to use an EFS volume.
If using the Amazon ECS optimized AMIs (so Red Hat), you'd have to install on the EC2 instances the package nfs-utils to handle that file system.
You can then create a /mnt/efs directory, and in the instance /etc/fstab file an entry:
<YOUR EFS CLUSTER DOMAIN NAME>:/ /mnt/efs nfs4 nfsvers=4.1,rsize=1048576,wsize=1048576,hard,timeo=600,retrans=2,fsc 0 0
In your service definition, you'll be able to declare volumes such as /mnt/efs/myvolume which will be shared accross your EC2 instances.
Then run sudo mount -a to activate this new network volume.
And because you don't want to do it on each created instance, you can just add this bunch of code in the user data section of the EC2 instance creation screen (or Launch Configuration creation screen in the case of an auto-scaling group). Don't forget to edit with your own EFS cluster domain name.
Content-Type: multipart/mixed; boundary="==BOUNDARY=="
MIME-Version: 1.0
--==BOUNDARY==
MIME-Version: 1.0
Content-Type: text/text/x-shellscript; charset="us-ascii"
#!/bin/bash
yum install -y nfs-utils
mkdir "/mnt/efs"
echo "<YOUR EFS CLUSTER DOMAIN NAME>:/ /mnt/efs nfs4
nfsvers=4.1,rsize=1048576,wsize=1048576,hard,timeo=600,retrans=2,fsc 0 0" >> /etc/fstab
mount -a
--==BOUNDARY==--
It will do all these steps for you at the instance creation.
If using restrictive security groups, you'd also need to add access to the NFS network port (2049 if I remind well).