I have tried the tarp suspension under the straps before and while it kept my tarp close to the hammock and remained taught the hammock suspension would get in the way.
Then tendertoe and rip waverly taught me to hang the tarp above your tree straps, around 8" higher or so. Leave some slack in your tarp ridge line and tighten down with your edge pull outs. It will give the tarp a slight natural catenary dip in the ridge. So far its perfect and very easy to adjust using prussiks on ridge line and pull out lines. I've never had trouble getting my tarp as low as my hammock ridge line and if I want it high I just loosen my edge tie outs and tighten the ridge line.
What a good idea. It does tend to be easier to tighten the side tie-outs than the main tarp suspension.