Smells like something is dead when I first get in...

Typical of a dead rodent . Probably in the heater box. You might try removing the cabin air filter (behind the lower glove box door) and check it to see if there is any nesting material present.

 

Could be mildew if the A/C runs a lot and the condensation drips and sits. After the next time, try running warm air without recirculation on for 5-10 minutes and it may dry it out and help.

 

A musty smell is usually associated with a plugged up heater box drain. That’s a little rubber hose that is located in the lower left, forward corner of the housing, right next to the center console. If it becomes plugged it can trap water. Stagnation then occurs and a related musty smell. If you go on YouTube you can watch all kinds of videos to help you with a plan on how best to go about unplugging it. It’s not rocket science, just a little inconvenient. You also mention that the smell deminishes shortly after the car is driven. That is probably because that the heater housing ventilates when it’s running...especially if you have the HVAC set on “Outside Air”.

 

have you put your nose to the stain area? what exact area was it?

 

try some baking soda

 

When you checked the cabin filter did you inspect under where it is located. Might need a flashlight and a mirror.

 

If the evaporator drain tube is plugged you may want to pull back the carpet under the glove box area and check for dampness and/or wetness. Sometimes the carpet and padding can get soaked from this condition.

 

I realize this reply is many years late, but just adding to it, in case someone else hits this issue. the drain trough under the Prius G3 windshield wiper cowl is poorly designed. this lends to debris build up and eventually poor drainage, which could back up into the cabin, via the cabin air filter. Also, as others have mentioned, mice and rodents can crawl in this space under the cowl and cohabitate with their earth-friendly, soybean-based plastic parts, which Toyota uses in some areas.

I have come to conclude that this cleaning should be on a regular maintenance schedule of 30k-60k miles, depending on where you live. Toyota never mentions it anywhere, which is another shame on Toyota, but I have cleaned these out on my cars, and they were simply gross. inch+ layers of organic matter and debris where rotting in there. it felt so good to remove the cowl and vacuum and flush that crap out of there.

here are some instructions:

removing prius gen three windshield wiper cowl - Google Search

I'm also attaching some pics of the dirt in my 2013 (@ about 60-65k miles); and mouse-proofing mesh I installed.

 

This only happened after you cleaned the trunk with dish soap? Most likely mildew. Get a charcoal bag absorber like this and leave in your car. https://a.co/d/1aLlnCV