So note, that all this advice is coming anecdotally, but hedged by some good science, I believe. I'm by no means an expert(nor do I claim to be offering that level of advice), but my undergraduate degree is in Biophysics, and I'm a high school coach. Also, all of my advice is going to be built on the assumption that your symptoms of shin pain are low in the spectrum of bone pain, that is, you haven't overloaded your bone long enough to put your bone stress injury into the realm of what would be in the stress fracture territory.
As someone who has battled "shin splints" for years, I've come upon various ways to troubleshoot them, especially when they pop up. To develop any plan of action of dealing with shin splints, first understanding what is going on is of deep importance. Without a precise understanding of what the problem even is, any attempt to solve the problem can't be systematic, repeatable, or approached in a knowledgeable way. With that in mind, it is important to understand that what shin splints are. They are a bone overloading problem, and the pain/symptoms felt are a result of the body's response to that, which includes inflammation (from my best understanding).
I firmly believe that that inflammation response is not always in the body's best interest. I think biology is erratic and the body does not always help itself out by functioning in the most optimal way, so sometimes that inflammation can be treated and help in a meaningful way to relieve symptoms (see "Stopping the Spread of Misinflammation" By Dr. Jonathon Sullivan). I have found success with compression sleeves, ice baths(which provide cold and graded compression), and even taking an NSAID when there are large flare ups. None of those things are "cures" but they can make the symptoms manageable.
From a biomechanics perspective, I have found that one of the things contributing deeply to my own shin pain is what I'll call a "lagging leg". Active forces on the tibia are the primary forces on the bone (bending forces) and if the talocrual joint (ankle) moves through a greater range of motion, the forces will be higher. One of the form cues I use is "snapping" my feet up earlier in the stride. This makes the leg come up with the foot in less dorsiflexion, and also has the added effects of increasing stride rate, which lower the forces during each foot strike (if the same speed is maintained, by lowering ground reaction time) and from my experience triggering something called the "stretch reflex" as well, which makes the calf more activated and could support the hard structures of the leg.
I would be careful in implementing this, and as with any form change, don't force your body to do anything. I would play around with how it feels, and search out to see if something like a "lagging leg" could be contributing to undue bone stress.
But ultimately I think you should keep in mind how slow bone actually adapts, and how increasing mileage is not a project to be dealt with in a few weeks, but rather months and years. I would also recommend to take a look over at John Davis's "running writings.com" which have a lot of useful information he has conglomerated on shin splints/medial tibial stress syndrome.
~Best of luck, hope some of this may have helped~