Hey there! I unfortunately cannot give any picture examples as I am at work right now, but you said you have the game save the player's progress.. when are you doing this? I have a couple ideas for this.
1. Save the game after each level. Then they would be taken to the "level" select screen for example.
2. You can save the game at any time, but have a "on save loaded" event so that when a save is loaded after starting the game again, switch to your main menu layout immediately. This would require you to have some global variable for keeping track of unlocked levels (which I assume you do).
3. Though I have never experimented with it, I'm sure you could store data about level progress by using "local storage" rather than a save-state, but these are my ideas. I wouldn't mind testing these when I get home if you would like me to.
Hope this helps!