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.
National Electrical Code says that the receptacle must be within 3' of the outside edge of the basin, but not more than 12" below the top of the basin (210.52(D)). It also states that the receptacle can be installed in the countertop, but must not be in the face-up position (406.5(E)).
Obviously the receptacles must be GFCI protected, since they're in a bathroom (210.8(A)(1)).
See also this answer from @Shirlock Homes.
National Electrical Code 2014
Chapter 2 Wiring and Protection
Article 210 Branch Circuits
210.52 Dwelling Unit Receptacle Outlets.
(D) Bathrooms. In dwelling units, at least one receptacle
outlet shall be installed in bathrooms within 900 mm (3 ft)
of the outside edge of each basin. The receptacle outlet
shall be located on a wall or partition that is adjacent to the
basin or basin countertop, located on the countertop, or
installed on the side or face of the basin cabinet. In no case
shall the receptacle be located more than 300 mm (12 in.)
below the top of the basin. Receptacle outlet assemblies
listed for the application shall be permitted to be installed
in the countertop.
210.8 Ground-Fault Circuit-Interrupter Protection for
Personnel. Ground-fault circuit-interrupter protection for
personnel shall be provided as required in 210.8(A) through
(C). The ground-fault circuit-interrupter shall be installed in
a readily accessible location.
(A) Dwelling Units. All 125-volt, single-phase, 15- and
20-ampere receptacles installed in the locations specified in
210.8(A)(1) through (10) shall have ground-fault circuitinterrupter
protection for personnel.
(1) Bathrooms
210.11 Branch Circuits Required.
C) Dwelling Units.
(3) Bathroom Branch Circuits. In addition to the number
of branch circuits required by other parts of this section, at
least one 20-ampere branch circuit shall be provided to
supply bathroom receptacle outlet(s). Such circuits shall
have no other outlets.
Exception: Where the 20-ampere circuit supplies a single
bathroom, outlets for other equipment within the same bathroom shall be permitted to be supplied in accordance
with 210.23(A)(1) and (A)(2).
Chapter 4 Equipment for General Use
Article 406 Receptacles, Cord Connectors, and Attachment Plugs (Caps)
406.5 Receptacle Mounting.
(E) Receptacles in Countertops and Similar Work Surfaces.
Receptacles shall not be installed in a face-up position
in countertops or similar work surfaces.
NOTES:
As @ArchonOSX points out, there must be at least one 120-Volt 20-ampere circuit supplying the bathroom small appliance branch circuit. This circuit can have no other outlets, unless it only supplies equipment in a single bathroom and any other outlets are in accordance with 210.23(A)(1) and (A)(2). (See 210.11(C)(3) above)
Best Answer
Given that the line cable (hot and neutral) enters the switch box, you can run a new cable from the light switch box to a new receptacle and connect it so the switch will still switch the existing receptacle.
You cannot run a new cable from the existing switched receptacle to get an always on receptacle.
If you wanted to do away with the switched receptacle, you could remove the switch from the circuit and connect the wires with a wire nut. The existing receptacle would then be always on.
You could install a ceiling box and light fixture and run a cable from the switch box to the light fixture. That way the switch would control the ceiling light.