Which site do you log into your hotmail account from? Is it something other than <
https://login.live.com/>?
You might try deleting your cookies to see if this helps solve the problem.
1.) Open, your preferences.
2.) Click on Security, click on "Show cookies...".
3.) Do a search for hotmail, select all the cookies that come up and Delete.
4.) Do a search for Live.com select all the cookies that come up and delete.
5.) Do a search for passport select all cookies and delete.
This should remove most cookies relating to the hotmail login but please note that this may cause other passport enabled sites to loose login information as well as hotmail.
Now try logging in again and see if the problem continues.