The cabling currently in your home is called daisy-chaining.
What you are proposing is all fine except that I would not attempt to make up the patchcord unless you have all the proper materials (correct RJ heads, crimper, tester, etc). The last 7' or so that the patch cord will cover is very small compared to the overall length of the circuit.
Your modem speed is dependant on a number of factors including the rate of pair twist in the cable, the lower the twist rate the more cross-talk [i.e. noise] that will get onto the telephone pair. Modem speeds decrease when the noise-to-signal ratio increases. While cat 6 is great for this, it will only solve the issue for the distance from the NIC to the modem. The balance of the cable from the Central Office to the NIC will play a bigger part as it is not unusual for this to be several miles.
Joseph Golan, RCDD