Unfortunately, the box for that receptacle is already full to the brim with wires -- 6 14AWGs is the maximum for a 2.5" deep single gang box such as the one you have here, and with the double allowance that is made for the receptacle, you already have too many wires in it as-is. Going to a deeper box won't quite get you enough space to tap here, either -- a 3.5" deep single gang has room for only 9 14AWGs without a device or 7 with one, and you need 8 14AWG wires (2 existing blacks, 2 existing whites, 1 existing red, 1 EGC allowance, 1 new black, and 1 new white) + a double allowance for the receptacle device.
So, you'll have to replace this box with a bigger box if you wish to tap the circuit here -- a 2 gang box is the simplest replacement option, with the other gang blanked out to make room for all those wires. Once you've done that, you can connect the new cable's black to either the existing blacks or the existing red, and the new cable's white to the existing whites -- you'll have to pull them out of the backstab and make a white pigtail to the outlet neutral, though. You'll want to put a ground pigtail on the existing receptacle and connect it to the existing grounds, in addition to tying the new cable's ground in, too.
The other option would be to remove the switch entirely from the the box that housed the switch that used to control this outlet (if you haven't done that already, that is), and then you can run the cable for your new outlet to the former switch location -- that box, sans switch, will have enough room to house the 6 14AWGs (existing black, existing red, existing white, EGCs, new black, and new white) provided it is the same depth as this one, and box replacement is an easier option in this case as you can go up to a 2.5" deep single gang if the existing box is too shallow.