I've setup a VPC on AWS and I'm trying to make one of my subnets private. I have:

1. Created my private subnet.
2. Created my NAT Gateway and associated my subnet to it.
3. Created a Route Table dest: 0.0.0.0/0 target: NAT Gateway
4. Created an Instance in my private subnet.

I can get to the instance in #4 by going to a different instance with a public IP then SSHing to its private IP. Once on the instance it has no internet connectivity.

Am I expected to update the route tables on the instance anything?