I can't imagine any chariot getting great MPGs with 1000 lbs of lumber in it, but is there anything that might squeak out a better MPG without too much hassle?
There are plenty of trucks that get better fuel economy, but they also tend to be far newer, and rather radically more expensive. Though when you get to large trucks, 1000 lbs isn't that much. I don't think I could tell you the difference in fuel burn with half a ton in my truck bed vs empty. Ride quality, drastically improved. Fuel burn change, minimal.
The problem is that if you're putting reasonably few miles on a truck (and putting the rest of the miles in a car), the difference in fuel burn just doesn't make enough of a difference to justify the rather drastically increased costs of a far newer truck. Going a few years newer usually won't make a difference, and going a decade newer costs a small fortune.
At $200/mo, $4/gal, 13mpg, that's about 650 miles a month, 7800 miles/yr. Go up to 20mpg (which is hard to find in a heavy truck), you're at $130/mo, which is $70/mo or $840/yr savings... but you've probably spent $15k+ to make the trade. It just doesn't make a lot of sense, financially, if you've got an older truck in good condition.
Get your heaviest stuff delivered to job sites directly, bypassing your need to haul them in your truck like suggested below?
Delivery is nice, but I've found, often enough, a large truck gives options that a small truck doesn't, in terms of being able to go get things cheaply, and having to rely on delivery services is... sometimes a problem. I
could get lumber delivered to my place for projects, but if I go and get it myself, I can ensure that I'm getting straight, non-mangled boards. If any exist.
For the rest of its life with you, can you do some tricks that might get you slightly better MPGs? Like when you need new tires, try ones that might offer better fuel efficiency?
Not too many options in large truck tires, and while I wouldn't run off road swampers on a work truck, I might not go with pure highway tires either if I were getting onto muddy jobsites regularly. My truck runs medium commercial truck tires (19.5s), and even with 4WD, my mud/snow traction is pretty poor. Toss a trailer on the back, and I run out of traction long, long before I run out of 4-Lo torque.
I would expect, from experience, the financially optimal option here, by far, is to keep the large, well maintained truck, only use it when needed (and be realistic - if your jobs don't require a large truck that often, consider a smaller truck or a van), and run it as long as possible.
This forum has a lot of weird blind spots about trucks, and Pete's opinions about trucks are valid for his use cases, but they miss an awful lot. There's nothing wrong with a huge truck if you use it as such regularly, and there are plenty of things you can do with a large truck that a smaller one just won't consider.