Splice the wires, making sure that whatever wire in the new piece is connected to a color on one end is connected to the same color on the other end. Leave plenty of slack.
How you splice is up to you -- for short term you can just twist the "stripped" (insulation removed) wires together tightly and wrap with some sort of tape (but such a splice will pull apart easily). For a robust connection you should use a connection block or crimp connectors and anchor the cables on either side so they won't be pulled loose.
(Cat 5 is used in place of phone cable all the time, so the cable is up to the task.)
(PS: With no internet connection, how did you post the above and how will you read this?)