So if im correct this is how your network looks:
If thats the case your router should have a default route to the inside interface of the ASA.
For all the networks connected to the switch you would have to make static routes (or use a routing protocol) that point to the interface of the router.
For example:
route inside 10.10.10.0 255.255.255.0 192.168.100.253
The 10.10.10.0 network here is a network connected to the switch. Wich the router also has a route for.
I only have one question tho. Why do you use a router ? It's better to have you switch route (given it is a layer 3 switch). Letting the switch route will reduce the amount of hops a packet needs to take wich results in a lower delay and less network traffic.
Currently working as Cisco Engineer at Neon-Networking.
Certifications:
CCNA - Have it
CCNA Security - Have it
CCSP - Almost!!!!
CCIE Security - Not so far away dream