Yes! When you upgrade your plan the billing month starts from the day of your upgrade. All unplayed games of the previous month will be transferred into your new month.
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For example, you played 2 of the 3 games of your Lite Plan and you realise Lite is not enough for you and decide to change to “Smart”. The one unused game will be transferred into your new first month of the Smart Plan, which means you get to play 7 games instead of 6 for this month.