There are several methods to skip the Gerudo Card as an adult to head to the Spirit Temple. Note if you would like to skip this as a child, check out the Spirit Temple Early page.
If you have the hammer you can perform this method. Get captured so you'll be thrown in the prison. Hookshot up to the ledge and aling your back so it's facing the gate. With the wall in front of you, perform a Hammer Boost. Hold the joystick in the upper right direction and at the end of the hover do a jumpslash. If all was performed correctly you will land on the hill. Proceed to the other side by hovering through the wall, it is not solid from behind. Be careful not to be too close to the guard because you can still be caught.
Get captured by the guards so that you'll be thrown in prison. Hookshot up to the ledge and align your back so that it's facing the gate. Now execute a superslide towards the ledge on the right, and equip the hover boots just as you are about to fall off. Here's the tricky part. Let go of R and move the control stick to right/up - right and roll. When the hover boots wear off, use a jumpslash to land onto the hill. Now proceed over to the other side of the gate and to the Wasteland.
Once you are up on the window ledge, pull out the hookshot and aim at the pictured black smudge on the wooden oning. You should be able to clip through onto the roof. This trick is useful when setting up the next two methods of skipping the Gerudo Card.
While above the jail, hop onto the wooden box. Face torward the wall next to you and Z-target. Equip the Hover Boots and sidestep off the box while still holding Z. As the hover wears off, sidehop jumpslash to the small banister. Run torward the steep hill and sidehop jumpslash. Carefully proceed to the Wasteland.
Make your way to the top of the jail. Take out the hookshot, Z-Target, and backflip so that you land on the side of the small fence. Shoot your hookshot during the backflip to cancel the flip. Now sidestep off torward the toward the taller ledge to grab onto it. Align yourself torward the slanted hill and megaflip to it. Now proceed over to the other side of the gate and sidehop off the hill.