<additional solenoid on the starter?> - yes.   That's actually the MAIN  switch that energizes the starter motor/engages the pinion.  

a lot of  'wont start/engage starter'  questions need to be clarified- 
- is the starter spinning,  but the engine isn't  being turned over?   Typically it's a bad starter drive sprag, or sticky shifter mechanism.
-dead silence when button or key is turned?  Could be a bad  coil on the starter solenoid,  or up on the firewall, or broke command wire, bad button/key switch... Most starting systems use  at least one additional 'relay'  between the starter button and the starter's main switch.

-just a loud click from starter when button or key is turned?     Could be burnt contacts in this main switch..  or a dead spot in the starter winding/brushes.