No, the Monthly Game is generated by your play on the Daily Games, once midnight on the last day arrives it’s then a new month, so it resets, and you start again.
Your picks cannot be carried over to a new month. We're afraid that this is not something that we would look to compensate for either. Not only is it free content with no monetary value, we also have no way of knowing what you would have picked. You may have won something or you may have won nothing.