Sean, great NY style pizza is about what I call the 3 H's- heat, high gluten flour and hydration.
Heat is, by far, the biggest player in the great pizza equation. High heat = quick baking time = great oven spring = airy/puffy crust. The stone is the vessel through which that heat is transferred. Thicker stones hold more heat and more conductive stones transfer that heat faster to the pizza. You can either shorten your baking time by modifying your oven to make it go above it's rated temp, by using a broiler technique and/or by using a thick conductive stone. I took a few minutes and looked at a few photos of Lombardi's pizza's, and, first off, I have to admit that consistency is not their strong suit. They seem to range from a relatively evenly colored 600ish degree NY style pie to a 750ish very slightly leoparded Neo-NY coal pie. Compared to the other coal places (Patsy's, Grimaldi's, Totonno's) this puts them on the low end of the temperature spectrum.
Before you do anything, you need to answer the following question: Which of the two Lombardi crusts at the bottom of this post are you trying to recreate? If it's the latter, then things get a little trickier. If it's the former, I'm relatively certain you can achieve this with the right stone. 1" cordierite kiln shelves are excellent, but, being a resident of Northern NJ, you have access to inexpensive soapstone slab, which is, imo, the best baking stone you can get.
For a higher coal-ish temp scenario, you could, in theory, just keep the stone you have and go with a cleaning cycle mod. Having re-wired ovens, I'm not a huge fan of cleaning cycle mods. If I were striving towards Neo-NY coal temps, I'd probably combine soapstone with either a broiler technique or a frozen towel cover on my thermostat.
High Gluten Flour
Another benefit of living in the pizza mecca of Northern NJ is that you have easy access to great flour. Corrado's in Clifton, Restaurant Depot, Dawn Foods in Edison- all have great pizzeria flour. These flours are all suited towards NY style, but for Lombardi's higher heat pie, I think they'll fit the bit as well. Maybe. You may need to do a blend of pizzeria flour and an unmalted flour like Caputo. Caputo is a little hard to find, but we'll cross that bridge when we have to.
Hydration is really just about shooting for a little bit above a flour's rated absorption value. For a classic bromated bleached 14% flour such as All Trumps, that translates into a 65-68% hydration territory. With the proper burst of heat, water will turn into steam and steam will do it's oven spring magic.
You've got a digital scale, right? How about an infrared thermometer? For where you're going, both are essential.
As far as the recipe goes, the Lehman dough is pretty much the standard approach. I'd pick one with some oil and some sugar. Both are pretty standard NY fare. Lombardi's may use a sourdough starter (Patsy's does), but, for now, I'd cut your teeth on a straightforward IDY recipe. Once you've mastered that, then you can graduate to sourdough. You'll also want to go with a dough with an overnight refrigerated fermentation.