An inch is a LOT of LME to leave behind. Follow the advice above to get the most you can out of the can.
There are many ways to aerate, and shaking is one of those ways. You can use oxygen and a stone (search the forums, you'll find more on this). The method I used last time was a 28" paddle attached to my drill. I let that whisk around for a minute or two while altering the depth I was stirring at. That seemed to do a pretty good job but I can't measure the actual oxygen content so it might be all froth and no aeration.
As for "the wort mixed well after doing a partial boil" - so far I have done partial boild ranging from 1.5 to 4 gallons. When you are done with the boil you usually need to add it to cold water in your fermenter. This is where you want to make sure the wort and the cold water are well mixed to get better readings.