UART to RS232 9 pin

Hi! I need to connect Arduino UART to the modem using full 9 pins set of RS232 signals.
I use levels conversion chip like MAX232, tried to connect DSR (6) to DTR (4) RTS (7) to CTS (8) like advised here: Implement a DTR pin for RS232? - #3 by system
this doesn't work.
Modem itself works for sure, I tried to connect it to notebook via USB-RS232 9 pin cable and it works. USB-RS232 3 pins (Rx Tx GND) doesn't work.
So the question: is there any way to convert 3 pin UART (or RS232) to 9 pin RS232?
Or I need to use spare digitals for emulating all remaining 6 signals?

Maybe this webpage will help you out?

Thanks! But it looks like one of the way to realize short-cut between pins 6-4 and 7-8

Maybe I misunderstood your initial post. I took it that you wanted to control the handshake lines, by if you dont, then, Yes, you can put loopback links on the handshake wires.