You won't likely stop them biting through unless the fabric is quite thick, and then it might not be as breathable for your needs.
You need to make sure there is enough space between you and the mosquito net (including where it hits the hammock). Either tie it up better, or add more strings to pull it away from you. Or, depending on your setup, add an object (e.g. small sticks) inside between you and the inner net to push it away from you.
If the only problematic area is under the hammock, then you could attach a thick blanket to the underside of the hammock, or lay it on the hammock, if this is comfortable to you.
You can buy better quality nets, which might be an option.
That said, you can get nets pre-treated with repellents, and are much more effective at repelling mosquitos than non pre-treated ones.
Such nets can actually kill mosquitos upon contact of the net.
And you can apply repellent quite liberally, how much depends on the net you have really.
But sounds like the problem is the net position, not net quality.