Using a 4-inch hoop for a 6-inch embroidery design on 8-inch fabric is possible, but it may present a few challenges. Here are some things to consider:
A smaller hoop can give you more control, allowing you to focus on small sections on the design at a time. Along with that, it is also lighter and easier to hold for long periods.
However, a hoop that is smaller than your design means you'll need to keep repositioning it. This can sometimes lead to uneven tension in your fabric would could distort your stitches.
There is also a risk of damaging the stitches you have already worked when you move the hoop, especially when you have dense stitchery.
If you decide to go ahead and use a smaller hoop try to plan your moves first. Make sure you have finished one section before moving the hoop to another.
You might also find a larger piece of fabric is useful, so that wherever you place the hoop there is enough around the outside for the hoop to grip and hold it securely.
Whatever you decide, enjoy the embroidery process.