Avoiding a greasy look with any oil, including this argan oil, depends entirely on the amount you use and where you apply it. The product's description notes it's popular for hair when lengths feel brittle or dull and need an extra layer of nourishment, and that it forms a light smoothing film around the hair shaft which can improve softness, shine and manageability without the heavy feel of some richer oils.
The key for fine hair is to use a very small amount—start with just 2-3 drops. Warm the oil between your palms and then apply it only to the mid-lengths and ends of your hair, which are the oldest, driest, and most porous parts. Avoid applying any oil directly to your roots or scalp. Focus on areas that are prone to dryness, split ends, or frizz. For a treatment, you can apply a slightly larger amount to damp or dry ends before bedtime and wash it out in the morning.
You can also blend a drop or two into your regular conditioner or hair mask for added smoothing benefits during washing. Since this is a cold-pressed, single-ingredient oil, it has a straightforward composition, but its effects can vary. If your hair still feels heavy, reduce the amount or frequency. This type of cosmetic oil is chosen for its simplicity and flexibility, so adjust the use to what works for your hair's texture and routine. External use only, and discontinue if irritation occurs.