There are several methods to skip the Gerudo Card as an Adult to head to the Spirit Temple. If you would like to skip this as a Child, please read the Spirit Temple Early page.
If you have the Megaton Hammer you can perform this method. Get captured so that Link is thrown in the prison. Hookshot up to the ledge and align Link's back so that it is facing the gate. With the wall in front of you, perform a Megaton Hammer recoil boost. Hold the joystick in the up-right direction and at the end of the hoverboost 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 Link is thrown in prison. Hookshot up to the ledge and align Link's back so that it is 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 is 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 Haunted Wasteland.
Once you are up on the window ledge, pull out the Hookshot and aim at the pictured black smudge on the wooden awning. 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 toward 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 toward 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 the Hookshot during the backflip to cancel the Link's momentum. Now sidestep off toward the taller ledge to grab onto it. Align yourself toward the slanted hill and megaflip to it. Now proceed over to the other side of the gate and sidehop off the hill.