The reason why the browser is not submitting cookie can be many reasons but when Internet Explorer is involved you need to check if all those sites are registered in the same Zone.
In the above use case, it is highly likely that site1 and login site are registered in the "Local Intranet" or "Trusted Sites" zone while site2 is "Internet" zone or not registered in any sites at all.
IE maintains cookies based on the Zones and do not share the cookies if the zones do not match.
site1 and login site are in Trusted Sites zone so the SMSESSION cookie will be submitted if the cookie was set in this zone.
If site2 is in a different zone, even if the cookie domain and path and secure flag match, the cookie would not be submitted.
Following article describes in more detail.https://blogs.msdn.microsoft.com/ieinternals/2011/03/10/beware-cookie-sharing-in-cross-zone-scenarios/