Your ADSL modem has router software as part of it's firmware. This is why the router gets an internal IP address from the modem that seems to make it just another device.
So, what you do is change the IP subnet that the Linksys router passes out, to coincide with the IP subnet that the modem passes out. Make the Linksys router just an AP point. This means that if your Modem will pass out more than two or five IP addresses, then it can handle the DHCP for your network. Otherwise, you can set up the Linksys router to handle DHCP for a range of... I dunno... everything above 100 (192.168.1.100 and up) but the gateway would STILL be 192.168.1.1. That would be the modem.
This means that all your connected computers would be on the subnet of 192.168.1.x and you would be able to connect to the Linksys router by connecting to 192.168.1.2 and the modem by 192.168.1.1 without having to manually change a thing.
EDIT because I have to assume if you are asking for a solution to avoid having to manually change your address, that you are currently having the Linksys router pass out a different subnet... meaning all the devices connected to the router are getting internal addresses of... I dunno... 192.168.2.x or 192.168.3.x or 10.0.0.x or whatever. The point is, you can change that subnet range that is being passed out from the Linksys router to match what is being passed out by the modem.
EDIT 2 Ok. From what you said, you should essentially be all set and good to go... except that your phone isn't getting an IP address. First... the Linksys should be connected to the modem in one of the standard LAN ports, as opposed to the Internet port that is on the router. From the initial basic setup page that you see when you first connect to the router's admin panel, you should be set up for Automatic Configuration DHCP. The Local IP address should be set to one that it normally gets from the modem (192.168.1.2 should suffice). Now. if you want the modem to handle passing out the IP addresses, you should have DHCP disabled from there. But again, the modem might only be configured to pass out a small number of IP addresses, which could explain why the phone can't get one. So, to configure the router to pass out IP addresses, you could Enable DHCP from that page in the router, set the starting address to be something like 192.168.1.100, and a Maximum number of users to be 50 (that should be default). If you Enable DHCP, the router wants some DNS numbers. Well, you can put 192.168.1.1 as the first one, since that would be telling the router to use your Modem for DNS. You could also put in others, like Google DNS (22.214.171.124 and 126.96.36.199) or you could get the actual DNS that your ISP uses (from their support website). But that should enable your router to pass out IP addresses as well.