I Had a Half Syringe of Restylane Under Each Eye with No Results. Will another syringe be helpful?

Restylane plumps up tear trough hollows and the amount required depends on the patient.  You should compare your results with your before photos to see if there really isn't a change in fullness.  You may require more Filler.  Restylane will not fade dark circles if they are caused by pigmentation.  Your physician may suggest a tretinoin cream and perhaps hydroquinone cream to help fade the pigment.  Wear sunglasses outdoors to prevent further darkening.

Dark circles under the eyes can be the result of several causes.  Truly, only one of the causes is treated satisfactorily with fillers.  Dark circles are generally the result of 1) Deep set eyes or a sunkeness of the lower eyelids that result in a shadowing effect, 2) darkening or hyperpigmentation of the skin (either hereditary, post inflammatory, or chronic inflammation), and/or 3) discoloration secondary to the vasculature underlying the thin eyelid skin.  If your dark circles are the result of deep set eyes or a sunkeness of the lower lids, then Restylane is an excellent choice to fill this area (a concave surface).  Furthermore, if this is indeed your case, 1 syringe per lower lid is completely reasonable.