Since the 1999 NEC, if you modify this branch circuit to extend it then you need to upgrade it to a properly grounded NEMA 14-30 outlet or it will not meet code. Most AHJ require an electrical permit to wire this extension and what you propose will not pass inspection.
From 2011 NEC below with my emphasis. Note that "grounded circuit conductor" is NEC-speak for the white neutral wire in your existing installation.
250.140 Frames of Ranges and Clothes Dryers.
- - Frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be connected to the equipment grounding conductor in the manner specified by 250.134 or 250.138.
Exception: For existing branch-circuit installations only where an equipment grounding conductor is not present in the outlet or junction box, the frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be permitted to be connected to the grounded circuit conductor if all the following conditions are met.
(1) The supply circuit is 120/240-volt, single-phase, 3-wire; or 208Y/120-volt derived from a 3-phase, 4-wire, wye-connected system.
(2) The grounded conductor is not smaller than 10 AWG copper or 8 AWG aluminum.
(3) The grounded conductor is insulated, or the grounded conductor is uninsulated and part of a Type SE service-entrance cable and the branch circuit originates at the service equipment.
(4) Grounding contacts of receptacles furnished as part of the equipment are bonded to the equipment.
Note that the grounding wire can be a retrofit and it doesn't have to be a home run back to the service panel. For example if you have a common grounded 110V outlet (NEMA 5-15R) nearby you can bond to this ground if the branch circuit goes back to the same service panel.
Here's a snippet of NEC that talks about retrofit grounds:
250.130(C)(1) A non-grounding type receptacle can be replaced with a grounding-type receptacle if an equipment grounding conductor is installed and connected to any accessible point on the grounding electrode system.
250.130(C)(2) A non-grounding type receptacle can be replaced with a grounding-type receptacle if an equipment grounding conductor is installed and connected to any accessible point on the grounding electrode conductor.
250.130(C)(3) A non-grounding type receptacle can be replaced with a grounding-type receptacle if an equipment grounding conductor is installed and connected to the equipment ground terminal bar within the enclosure where the branch circuit originates.
250.130(C)(4) A non-grounding type receptacle can be replaced with a grounding-type receptacle if an equipment grounding conductor is installed and connected to an equipment grounding conductor that is part of another branch circuit originating in the same enclosure.