T-100 balks at cold start

[T-100&Tundra]

My 3VZE 6-cyl engine in '94 T-100 DX started "clicking" instead of turning over last fall. Replaced battery and now everything else electrical works, but in the mornings it still "clicks" instead of turning over until I try it a few times. Have checked connections to battery and they appear okay. Could this be a solenoid that is getting tempermental to cold? After these issues with the first start of the day, no further problems are encountered with starts for the rest of the day.

Does anyone know if the solenoid on the starter can be replaced separately or is it made in one unit with the starter motor? Also, could something else be the issue besides the solenoid?

Thanks in advance for your assistance!

[xsacta]

clicky:

http://www.4x4wire.com/toyota/maintenance/starter/