You have a few options there :
1 - You actually want to pull the image after the DockerHub build is done, not after travis is done, in that case you can use DockerHub webhooks to call some kind of service (it can boil down to any kind of script) on your server, which will pull the image and run it.
From https://docs.docker.com/docker-hub/repos/#/webhooks :
You can use a Hub repository webhook to notify people, services, and
other applications after a new image is pushed to your repository
(this also happens for Automated builds). For example, you can trigger
an automated test or deployment to happen as soon as the image is
available.
2 - Use something like Ansible to avoid SSH'ing into the machine yourself. You write an Ansible role once, and it will run in your Travis build. There is a very good docker module for Ansible so you'll be good.
3 - Use a more adapted infrastructure like Amzon ECS, which is basically one or a group of EC2 instances that run and manage Docker containers and images for you. If you know how to spin an EC2 instance, making a simple ECS instance will not be a problem at all.
I hope this helps.