Generally, a gift subscription will not renew. That being said, if you wish to set up an auto-renewing subscription for someone, you just need to choose the “Automatically renew this subscription” option at checkout. (If you don’t see it right away, it will show up after you tick the “This is a gift” box.
With an auto-renewing subscription, you don’t need to worry about missing a box or taking any further action. The subscription will continue to auto-renew for another cycle until you decide to cancel.