By CSS, I assume you are talking about the Cold Start Valve (CSV) on the intake manifold? You say cold start solenoid, but that solenoid on the IP richens the IP during starting. It is activated everytime you start (depending on your model which you didn't state). You also mention the thermo-time switch which controls the CSV not the CSS. So there lies my confusion. I was thinking you were talking about manually energizing the CSV which is a common procedure. I have never heard of manually energizing the CSS and not sure it would do anything. It is helpful to know what year and model you have as these cold starting systems vary greatly.
By having to manually energize for 10 seconds the CSV, sounds like a long time to me. Is this prior to starting or during starting? When I manually activate the CSV prior to starting, the CSV also fires during starting as normal assuming the engine temperature is under 90*F or so. I have it wired so that the CSV can function normally or can be manually energized. However, I never energize mine for more than 3 seconds or so.
It sounds like you may need to diagnose if the Thermo-time switch is working properly which controls the CSV. You can also check the relay that is between the TTS and the CSV as suggested earlier. The TTS switch stays on relative to what the engine temperature is. Colder longer, warmer less. So if the ambient temp is 80F*, it may only stay on a second or two. I believe the cut off temperature is something like 90 to 95*F. So the closer you get to the cut off temp, the less the time it stays on during cranking. There is a section in the tech manual regarding testing the TTS.
Let us know if we are talking about the CSV or the CCS.