How are you giving a bath?

I am just looking to check what/how others are doing. Baby is 8 weeks old and we have the skiphop Moby smart sling tub. Husband and I bathe baby together.

We fill up the tub with warm water (we often also add BM) and let baby sit in, covered with a wet washcloth. We start by washing her face (no soap) and work our way down to her private parts, sudsing her up with a cerave baby wash/shampoo. We wash her hair as well.

This is now the part which i am confused if we are doing correctly or if it could be done more efficiently. As the tub water is now effectively dirty and we don't know if she peed in it either, we give her a second wash with clean water. I will hold baby in a heated towel for a few minutes and husband will quickly dump the dirty water and rinse out the tub, and refill with clean water. We put her back in and just rinse her body and hair over a few times, before putting her in a new clean towel and drying her off.

Sometimes she gets fussy (from being wet & bit cold) when i hold her while husband is changing out the water. How are you going about bathing?! Wondering if there's a better approach?

EDIT: Thanks so much for all your replies! As a first time mom, this gave me a lot of perspective! We did try giving her a second quick rinse with a bucket of warm water while holding her suspended over the dirty water tub or sink while she was a newborn, but she hated it and was thrashing and wailing. We tried this technique again this evening and it went smoothly, a cry free bath!! Definately saved us a lot of extra bath work. I also plan to try the other method of showering while holding her, sometime!