two routers on the same lan

In your first router, configure a route to 10.20.0.x via gateway 10.10.0.2. Make sure you have not turned off redirects.

All your client systems will send the traffic for 10.20.0.x via their default gateway 10.10.0.1 (your first router), which will send it on via its route to 10.10.0.2. But it will also send an ICMP redirect packet to the client, which will then send the next packets directly to the gateway 10.10.0.2.

This usually works quite well, but you have to beware that sometimes such redirect packets are turned off or filtered in a firewall. In that case all your traffic to 10.20.0.x will take an unnecessary hop via your first router.

Of course you could also set a fixed route to 10.20.0.x via 10.10.0.2 in all your client systems. But that may be a lot of work and when you change the network you may need to do it all over again.