In our experience, the more you can do to make the CRM your own, the more customization and work you put in to branding it specifically for your business the better.
With this in mind, why not set a specific landing page for your users who log out of their Salesforce interface.
You can set this to pretty much anything you want – it can take them to your company’s intranet perhaps or another Dashboard or portal that you have.
To achieve it just go to Setup and Session Settings. At Logout Page Settings, just paste the URL you want users to land on when they log out.