Signal to noise degradation based on noise temperature vs noise figure
In plain language, the problem is that you warmed your signal up without adding any gain.
Unless you actually chill that first stage to the same temperature as your source (it's been done), then if you don't add enough gain to overcome the noise temperature of 290K, your theoretic noise figure is meaningless.
Eg. if we run your math for a .8 dB noise figure relative to 10K, then we get a 2K noise temperature.
But your system runs at room temperature, so you followed standard practice and calculated relative to 290K, and got a noise temperature of 58.7K.  That's reality.  You then tried to figure out the output noise power, and came out to the truth for the temperature of your system, which is to say quite a bit worse than expected.
The conclusion is that if you want to actually see your theoretic noise figure, you have to add enough gain that the amplified input noise dominates over the thermal noise of the electronics (at whatever temperature they operate at).