Kitores of the morning was brought after the sprinkling of the korban tamid and the bringing of the limbs. In the afternoon, the kitores was brought in-between the bringing of the limbs of the tamid and the pouring of the libations.

If the cohen gadol was old or delicate, they would take hot water and pour it into the cold water of the mikva.

Yuma 3:6

They brought him back to the House of Parva which was in the kodesh. They spread out a linen sheet between him and the people. He washed his hand and feet and took off his clothes. Reb Mayer says first he took off his clothes and then he washed his hand and feet.

He went down to the mikva and got up and dried himself. They brought him the white clothes of the cohein and he dressed himself and washed his hands and feet.