I've been having pain on the sides of my feet (outside about midway back on the foot(. I was wearing a pair of Sidi's size 43. Went to a Specialized 44. Still have the pain, so I'm thinking it's something else, like how I have the cleats bolted on the shoes or even the pedals. How do I diagnose further?
Two major possibilities here. Could be a combination of both. I'm tempted to say it's the first one I'm going to talk about:
Given where you said your feet are hurting (outside, front) I'd say you have your cleats toed in too much. This is caused by your foot constantly trying to re-align towards what's neutral for you.
Here's what to do:
What you need to do is take your right shoe, loosen the cleat bolts, and with the sole facing you turn the cleat ever so slightly- just a couple degrees before re-tightening. Take your left shoe and do the opposite; turn the cleat counter-clockwise just a couple of degrees. See if this alleviates the pain at all. Repeat if necessary. You'll know you've gone too far if you start to get pain on the inside front or outside rear of your foot.
The other possibility is that the shoe doesn't fit you quite right.
It may be that the front of your foot is a little wide and you need a wider toe box. Specialized shoes do have pretty wide toe boxes up to their Pro line. The S-Works models get noticeably narrower in the front of the shoe. Despite the fact that you were fitted for a standard width, a wider shoe may be more comfortable. As long as your foot isn't sliding around side to side the shoe isn't too wide. It's important to note that your foot flattens/widens when you press down hard on the pedal so if your shoe is already pretty snug in the front you may need more room. The material that specialized makes their shoes out of (at least the uppers on their S-Works shoes) does not stretch very much, so if it's at all tight now it will probably remain that way.
Don't dismiss the possibility that the shoe may just not fit your foot well from the get-go, regardless of width or size. Cycling shoes are deeply a matter of personal preference, arguably falling right behind saddles. What works great for one person may be a device of torture for another. Keep this in mind and if the above suggestions don't help you, you might want to consider buying some cheaper model used shoes from different brands on ebay before forking over big bucks for a nice pair of shoes that you can't really try before you buy.
Shoe fit and the stiffness of the last are the first things that come to mind. I'm thinking you have a wide foot and cycling shoes are typically on the narrow side.
I've always had major issues with any shoe that was slightly too narrow for my foot. I'd get cramping along the outside edge of my foot. It gets worse when the shoe becomes stiffer (It's just hell for me to wear X-country ski boots or hockey skates). As cycling shoes are meant to be something akin to a block of wood, that means for me I need to be certain the shoe isn't too narrow or I'll be in pain really quick.
For my road shoes, the size is such that I actually have a slight amount of heel slippage which allows my foot to flex a bit more than what typically would be expected. With my Mtb shoes, I go for the freeride type of shoes which don't give nearly the same amount of energy transfer but my feet are way happier at the end of the day.
I had a pair of Sidi shoes a few years back (Awesome construction) but found that even the Wide width was narrow and caused me pain after a few hours of riding. Specialized seem to fit narrow as well. I've had to try on many a shoe to find a wide enough fit and usually I'm still left with having to go sockless just for a little bit extra comfort.
Get your foot sized by an expert and go from there. If it's a width thing you'll know with numbers. If it's something else then you can start to narrow things down.
How is the condition of your pedals and cleats? Maybe they are worn out and they need a replacement. I've had a friend who rode with worn out pedals and he had a lot of damage to his knee.
My cleats also weren't attached the right way at first. Someone who was specialized in body shapes on a bike attached them right on how I put power on my feet and pedals. Maybe you are giving too much power on your feet and your cleats aren't attached the right way either
(sorry for my poor language, I'm not native English)