I did the upgrading to Magento 188.8.131.52 with Magento Connect but after doing that still appear the message "October 27, 2015: New Magento Security Patch (SUPEE-6788) – Install Immediately".
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Magento CE 184.108.40.206 includes the fix of all security issues that are addressed by Patch SUPEE-6788. So there is no need to apply the patch once you have done a proper upgrade.
To take full advantage of all new security features go to
System > Configuration > Advanced > Admin > Security and set
Admin routing compatibility mode for extensions to "Disable". More details
Don't forget to check the customer registration forms for your store. If you use customer registration template files in your custom theme then make sure the
form_key is added to those files. More details
Also check whether all variables and blocks that your store is using are added to the white list in
System > Permissions > Variables and
System > Permissions > Blocks More details
And now the answer to your question...
The message you see comes from the notifications inbox. This message will not disappear when upgrading to Magento CE 220.127.116.11 (and also not when applying Patch SUPEE-6788).
You will have to do this manually (in both cases). Go to
System > Notifications and mark this message as read or remove it from your Messages Inbox.