You'll need to have it properly rinsed and extracted. Obviously the person who performed the work lacked the knowledge and ability to accomplish the task. Be thankful it was only a second hand mattress. There are many people in this industry that don't get the education they need to perform the work correctly. Unfortunately consumers view that a cleaning service as a commodity, its not. Family owned or owner operated services are your best bet. They have a reputation to uphold and it is in their best interest to do the best work possible. When ever hiring an in-home cleaning service, it pays to do the research before you hire them. Check their reviews online, ask them how long they have been doing it, experience, what type of cleaning agents (toxicity), do they have insurance and any other concerns you may have.
You post mentioned using a steam mop for the steam cleaning on the mattress. I'm not sure about that, but I can say that I have gotten large odors out of mattresses with a steam cleaning. I used an upholstery steam cleaner to do it though and put a fan on it. In some cases, children's night-time accidents, I have had to do multiple cleanses. But, it eventually worked.
I use Medicated Powder to absorb the smell, it is helpful to remove unpleasant smell, as well as powder is a safe option. You can also neutralize odors by using Vinegar and baking soda.
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