This documentation assumes that Exchange 2010 is outlined for client access. This way you will not encounter any error messages from your Elastic Email account or error in delivery as you should set it to be able to send emails to the public.

To know more about Send Connectors, you may check this helpful link from TechNet.

Send Connector configuration is pretty simple. This is connected setup that you may require to ensure that you'll gain access to connectors. Here's how:
1. Sign in to your Exchange server or a server that has the Exchange Management Console installed for administrative purposes.
2. Go to Start > All Programs > Microsoft Exchange Server 2010 and click on the Exchange Management Console item.
3. Roll open the Organization Configuration list and select Hub Transport.
4. Click on New Send Connector in the Actions sidebar.
5. You may name the Send Connector anything you wish to and select Internet for the Intended use drop-down. This allows for the integration with an authenticated SmartHost and adds the default send permissions required by Exchange 2010 to route messages through an external host that is not a part of the domain forest.
6. Click Next.
7. Click Add and specify the allowed address space to which the connector can send messages. If you want to send to anyone on the internet, then you would insert a “*” in the Address field. If you want to send only to a specific domain, enter that domain. If you want to allow all sub-domains you can check the box, but I prefer to notate that literally like so: *.SAMPLEDOMAIN.COM. Insert a cost's height (100 is the top, so choose wisely) so that the connector is not used inadvertently.
8. Click Next.
9. On the Network Settings page, select the Route mail through the following smart hosts item.
10. Click Add.
11. Select the Fully qualified domain name (FQDN) item, and enter: smtp.elasticemail.com
12. Click Next.
13. In the Configure smart host authentication settings page, select the Basic Authentication option and check the Basic Authentication over TLS checkbox, then enter your Elastic Email SMTP username and password.
14. Click Next.
15. If you only have one Exchange server, it should automatically appear. If you want to use another server(s) in your domain click Add to locate the servers you want to allocate to this connector.
16. Click Next.
17. Verify your settings and, when satisfied, click New.
18. If the configuration completes correctly, you will receive positive notification. You can then click on Finish.

Once you have completed installation of the connector, you will need to configure your server/s to allow limited access to the new send connector.
You can read more about Send Connector permissions.