This question really hits home for me as my home lake is weed choked in the shallows and I've had difficulty catching fish on a regular basis.
What I've done is fished my strong suit (off shore structure) first. I find that first and foremost, those fish are easier to locate. Secondly, they don't seem to be affected as much by slight changes in the weather like shallow fish are. Yes, I'll switch from faster moving hard baits to soft plastics after a front passes, but unless the barometer is really high, I'll stick with the deeper fish.
That's me. If you feel more comfortable fishing the mats, I'd say start there. Just know that those fish tend to be loaners and they will react faster to weather changes, often times suspending under the cover. Punch rigs are great at penetrating the cover, but many anglers use them strictly as a bottom contact bait and miss out on those suspended fish.
In short, IMO, unless you are proficient with fishing either, gravitate toward your strong suit. The exception for me is if a frog bite is on during pre-front conditions. Then, I hit the mats for some hand to hand battles.