Weeds are tenacious plants and find ways to overcome the barriers we place in their way.
I suggest manually removing the matting and as much of the weeds as possible.
When putting the new weed mat on top, make sure that the edge is pinned very well to the ground and that there are no holes or cuts in the weed mat that will allow the soil to be exposed to light.
Using two weed mats should be more than enough to prevent weeds from growing for some time.
But please keep in mind that eventually the weed mat will wear out and the weeds will find an opening to grow into. - EricL
My experience is dirt and seeds fall onto the weed mat and the weeds germinate on top of the weed mat sending roots down in through the mat by passing you mat's protection or growing around the edges from underneath.
You really don't want to spray poison near your plants. So I use a small 10mm wide artist brush and have a jar of weed killer at high strength. I simply paint a couple of the weed's leaves and they die in a week or so.
There's and old gardeners saying one year of seed is seven years of weeding. Don't let weeds go to seed. Pulling them out can also leave roots behind and you are wasting your time, plus it is a lot of bending which is not easy for everyone.
Tip: Add 1 to 4 drops of dishwashing detergent into your jar or spray bottle of weedkiller. It breaks the poision mix's surface tension and wets the plant and ensures you don't have drops that run off. Also make sure you don't poison when rain is due. - Jewelleryrescue
Weed mats are not designed to stop weeds altogether, but create a barrier to reduce the weed roots becoming fully established. It makes weeds typically easier to remove. Extra layers could be tried, but won't stop weeds altogether. Unfortunately where there's dirt, there's going to be weeds.
There used to be a Zero brand weed wand as an alternative to a spray. The poison soaked a wick and this was touched onto the weed so there was no overspray to affect nearby plants. Very effective especially for a small area or rocks. - kacharash4