Our "Navy" sealift is predominantly handled by Military Sealift Command. which is a mix of United States Navy Ships (USNS Whatever) and long term charters. The USNS ships are civilian crewed, and they tend to be more specialized use ships (Large Medium-Speed Roll-on/Roll-off, Expeditionary Fast Transports, Maritime Prepositioned Stocks, etc.) The MSC Long term charter ships are just normal cargo ships we have sole us of. They tend to be a mix of breakbulk cargo/container ships and RO/RO ships. The "civilian" ship you saw in Olympia was almost certainly one of those as they tend to be the choice for moving units Battalion and larger. Excluding oilers there are 33 USNS Cargo ships (Dry cargo or RO/RO) available to MSC and 20 of them are supposed to be Prepositioning ships (although they aren't all prepositioned at all times)
On top of that, SDDC (Surface Deployment and Distribution Command) has contracts with common carriers for when we need to move less than a ship's worth of stuff. So when your unit loads their office supplies up in 5 connexs in Germany and ships it back CONUS, it probably goes on a civilian ship just like normal sea freight.
How much stuff is moving at once, how sensitive it is, how much we are willing to spend, and how important the delivery timeline is all play a part into which mode of sea freight the US chooses to move stuff around the world.
If you are saying to yourself "Wow, does it make sense to have so much important logistics tied up and dependent on civilian ships and crews when we are looking at a near-peer conflict with countries capable of closing sea lanes?" you are not the only one,