For most of my cooking, I use olive oil, butter, or coconut oil. On occasion we'll have some lard, but I'm not too experienced with it. Coconut oil seems to be a good choice for high temperature cooking because saturated fats are more stable and it has a higher smoke point. Butter, while being composed of saturated fat, seems to burn easily, so it doesn't work well for frying. Cooking with extra virgin olive oil is not recommended due to its low smoke point and relatively unstable monounsaturated fats. I do have non-extra-virgin (usually labeled "light") olive oil for medium or higher temperature cooking because it has a higher smoke point, but I'm not sure how drastically the refinement process negatively affects the nutrition profile of the oil.
tl;dr: So, for high temperature cooking, right now all I seem to have is coconut oil. Any other recommendations?
I think the light version of olive oil comes from rotten olives that they could not use for virgin oil. They refine it to remove the smell.
Two alternatives for high temperature cooking is tallow or palm oil. I havent seen tallow in stores. Lard could be as high as 30% in PUFA.
You could try ghee (aka clarified butter) if you want a high temperature resistant butter. You'd slowly heat up a pound or two of butter until it melts, and allow it to simmer at a very low temperature to let the moisture out, then scrape off the stuff on top, then slowly pour through a paper coffee filter to get the rest.
Red palm oil also holds up very well to higher temperatures, but it has a strong flavor, so its use would be limited to things like curries.
And of course, as mentioned earlier, avocado oil.
You can also mix some oils together... i.e. ghee + avocado oil, or EVOO + butter.
Generally any low PUFA oil (higher in MUFA or SFA) will do just fine. Of course, you could cook at lower temperatures and avoid the problem - even sauteeing in EVOO works fine at low temps.