Will increasing the wing area result in higher aircraft range?
Up to a point.
There is an optimum for wing area that shifts with operational considerations and airfoil type.
Propeller aircraft benefit little from flying higher, and due to the high efficiency of piston engines, the highest wing loading possible with the desired take-off distance will give the best result. Aspect ratio should be as high as practical; past examples used values up to 14.5 for achieving high range.
Jets benefit more from altitude but need more fuel, so my main focus will be on jets. Now it helps to look at the Breguet range equation: $$R = \frac{v\cdot L/D}{g\cdot b_f} \cdot ln\left(\frac{m_1}{m_2}\right)$$
The indices denote the state at the beginning (1) and the end (2) of the flight. For a higher range $R$ it helps to increase

Fuel fraction $\frac{m_1}{m_2}$: The higher, the better. But note that doubling fuel will not double range. Bigger wings will allow to hold more fuel, so here wing size helps.
Flight speed $v$: This allows to cover more ground while in the air. However, flying faster will produce more drag and consume more fuel once you fly faster than at minimum drag.
Lift-to-drag ratio $L/D$: Obviously, producing less drag for the same lift is equally important. In order to shift the best L/D to higher speeds, a high wing loading helps because friction drag grows less than linearly with dynamic pressure.
Engine efficiency: Less fuel for the same thrust will let you fly longer. $b_F$ is the thrust-specific fuel consumption. Since this factor is independent of wing area, it can be disregarded for this answer. Indirectly, however, it helps to have more efficient engines because then the need for a high fuel fraction is less pressing. The inefficiency of their engines is the main reason for the low wing loading of the first generation of transatlantic jets.

An increase in wing area allows for a higher fuel fraction, as long as installed engine performance is sufficient to power the larger, heavier airplane. Both the higher mass and the higher wing area will increase drag, so the optimum range speed will decrease for increasing wing area. The tipping point where increasing area further will no longer yield an increase in range is reached when the fuel fraction is already so high that increasing it further buys too little range to make up for the lower cruise speed and the higher fuel flow enroute. For most configurations, the lower power loading of the larger, heavier airplane with the same engines will most likely set the upper limit of wing size.
The actual optimum shifts with details in the airplane's layout and operation. Supercritical airfoils with their higher relative thickness at transsonic speed and more efficient engines will directly help to increase range for the same wing area. The requirement for low altitude flight and not having to take off and land can also influence the result.
Don't neglect scale: Larger airplanes have disproportionally longer ranges than otherwise identical, smaller types.