The WP2Static S3 addon allows invaliding CloudFront distributions when deploying. An invalidation would make index.html (and other likely cached paths) immediately update when new content is posted.

I think it would take the following:

1. Update the ECS task execution role to allow invalidating the CloudFront distribution created by the cloudfront module.
2. Either:
   a. Document manual configuration changes for the S3 Deployment add-on (see also docs: add additional setup instructions #69); or,
   b. Automatically update the configuration for the user (see Additional pre-configuration of S3 Add-on: Activate and ACL #15)