No, the information that would appear on the 1096 is included in the E-File.
More Information about E-Filing Form 1096
The E-File is a combination of the tax reports you are filing, including but not limited to the 1099, 1097, 1098, 3921, 3922, 5498, and W-2G. This is a summary tax report used to compile the information from all reports to different employees and contractors. When you e-file something like a 1099, that form does not need accompaniment because the necessary information is already included.
Technically, there is no such thing as e-file 1096. Form 1096 is used to transmit paper forms to the IRS, but if those forms are filed electronically, 1096 is unnecessary. In sum, if you are e-filing your 1099 forms, you do not need to e-file 1096.