Endless Islands have no time limit, so you can explore the lands at your own pace. The portal leading a new island is located somewhere at 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 Endless 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.