Please check out this AWS Documentation for managing multiple ip.
Relevant part is that you need to assign a secondary private ip to your istance and then you can add a second elastic ip.
To assign a secondary private IPv4 address to a network interface
Open the Amazon EC2 console at https://console.aws.amazon.com/ec2/.
In the navigation pane, choose Network Interfaces, and then select the network interface attached to the instance.
Choose Actions, Manage IP Addresses.
Under IPv4 Addresses, choose Assign new IP.
Enter a specific IPv4 address that's within the subnet range for the instance, or leave the field blank to let Amazon select an IP
address for you.
(Optional) Choose Allow reassignment to allow the secondary private IP address to be reassigned if it is already assigned to
another network interface.
Choose Yes, Update.
Then
To associate an Elastic IP address with a secondary private IPv4
address in EC2-VPC
Open the Amazon EC2 console at https://console.aws.amazon.com/ec2/.
In the navigation pane, choose Elastic IPs.
Choose Actions, and then select Associate address.
For Network interface, select the network interface, and then select the secondary IP address from the Private IP list.
Choose Associate.
I try to avoid the need of more public ip on a single server and looks like AWS thinks that having "elastic ip" is generally not needed as you have just 5 of them by default unless you ask for more.