How to Replace Old 1/4" In-Line Copper Water Filter?
It appears to me that the copper tubing connected to the outflow of the filter is long enough to reach the saddle valve. ( upper right corner in Photo ? ) 

If that is the case you can eliminate the extra tubing feeding the inflow side of the filter and just go directly from the saddle valve to the copper tubing that is feeding the refrigerator.

You may be lucky and the threaded fitting on the outflow tubing will be the correct size and kind to attach to the valve.  

The saddle valve could be a 1/4" OR 3/8" connection. If you turn off the valve and then disconnect the line from the valve to the inflow side of the filter you can then mark the valve end and the filter end  of the tubing and take it to the store so you, or the associate, can determine the correct coupler fitting to use to connect your existing fitting to the saddle valve. ( the inflow threaded fitting appears to be the same as the one on the outflow side. )

If it were me i would replace the saddle valve, they are notorious for failing due to corrosion over the long term.  This require shutting off your water removing the saddle valve, cutting out the puncture from the saddle valve and soldering in a Tee, ( or use a push to connect fitting ) Then put a 1/4 turn valve on the tee branch for the refrigerator and the correct fitting to marry to the threaded fitting on your exciting tubing that goes to the refrigerator.