I use shampoo as well. Don't use a highly perfumed one.

Woolite is (or at least used to be) just linear alkylbenzene sulfonate in water. Any of the typical anionic surfactants will work and work well. Sodium lauryl sulfate or sodium lauryl ether sulfate are the ones typically used in shampoo or LDLs (light duty liquids, or hand dish detergent). For that matter, the only major difference between shampoo and hand dish detergent is the perfumes.

Liquid hand soap is not soap and could also be used. A bit foamy so may require extra rinsing. They are typically based on alpha olefin sulfonates and or betaines.

About the only thing you wouldn't want to use is machine dish detergent. Even liquid tide can be used in a pinch, very dilute. But if you are looking for mild it is best to avoid the products with alkaline "builders" (phosphates, sodium carbonate, etc)