If you can already access some server/computer remotely through VPN connection, you should be able to configure the ASA to include those other devices too.
If you can provide your ASA configuration (removing any sensitive information from it) we can go through the needed configurations steps for you to be able to directly use the VPN Client computer to access the remote devices with SSH.
If you post the configuration here, please also provide the IP addresses (or a network/subnet) of the devices you need to access
Did I undestand you correctly, is the situation the following
You connect to the company network from home with VPN Client
The Linux boxes are located behind a L2L VPN connection thats between the same VPN device that you connect to with the VPN Client
What you would need to confirm is
That the remote network(s) that holds these Linux box is included in the VPN Client Split tunnel configurations
If the VPN Client is configured to tunnel all traffic you dont have to do anything
You have to make sure that the traffic from the VPN Client pool is also allowed to the L2L VPN or if possible NAT the VPN Client users to some address that is already configured on the tunnel
There is also other factors to the this setup but I can only give some general advice without actually seeing your ASA firewall configuration. If you want you can post that for us to go through (remove any sensitive information like public IP:s, username/passwords, etc)