Sometimes Adventureland is closed in the mornings! However, if it is open, I would take the "passageway" on the right just past Aloha Isle (home of the Dole Whip). Then I would turn left in Frontierland and head for Splash.
A second shortcut would be past the Tiki Birds, on the right you will see the Sunshine Tree Terrace (it's a counter service juice bar type of place). You can also cut through there to Frontierland.