This issue happens as once the user logs out, the page can still gets served from the cache. Hence the cache needs to be cleared once the user logs out.please include below lines of code in your logout function to clear cache.
HttpContext.Current.Response.AddHeader("Cache-Control", "no-cache, no-store, must-revalidate");
HttpContext.Current.Response.AddHeader("Pragma", "no-cache");
HttpContext.Current.Response.AddHeader("Expires", "0");