I have an AWS instance running behind an elastic IP. The instance also has a private IP address in my VPC. I will be re-assigning the elastic IP to another instance and am tryin
Unless this is a 'secondary private ip address' (and you said it was not), then no, you can't:
You can assign additional private IP addresses, known as secondary private IP addresses, to instances that are running in a VPC. Unlike a primary private IP address, you can reassign a secondary private IP address from one network interface to another. For more information about primary and secondary IP addresses, see Multiple IP Addresses in the Amazon EC2 User Guide for Linux Instances.
http://docs.aws.amazon.com/AmazonVPC/latest/UserGuide/vpc-ip-addressing.html