Daily Adventure has no time limit, so you can explore the island at your own pace. The portal leading a new Daily Adventure is located somewhere at the end of your current island. If the portal is covered with fog, its whereabouts are highlighted with blue sparkles.
Once you find the portal, you can travel to a new Daily Island at any time. If you decide to do so, be sure to collect all the bonuses and resources on your current island as it'll close forever.