I cannot seem to reproduce this using the same steps.
How are you logging the user out ? Also with a custom widget ? What is the code? Did you also delete the authorization cookie?
To delete the .ASPXAUTH cookie created you can clear it by logging out using a call to /Sitefinity/Signout
Do you have the same behavior when you use/test the Default provider?
If the issue only happens with your custom provider, what I also can suggest is to open a support ticket as it may be required to send us your project data in order to isolate the reason for this issue.
Do you want to have your say in the Sitefinity development roadmap? Do you want to know when a feature you requested is added or when a bug fixed? Explore the Telerik Sitefinity CMS Ideas&Feedback Portal
and vote to affect the priority of the items