I am considering getting a new engine for one of my cars and I was wondering if I'll have to change the odometer too. Since the car already has 220,000 miles on it, it would become a huge number if I used the same odometer on a new engine.
I wouldn't touch the odometer. The rule I always subscribe to is that the odometer measures how far the chassis has rolled. Otherwise a person could go mad trying to figure out which repairs/replacements should reset the clock. (Obviously not tires or wheels, but wheel bearings? Axles/driveshaft? Transmission/differential? etc...)
It also makes sense when you think about the disconnect between the number of hours an engine has run vs. the distance it's traveled. Work trucks, limousines and police cars all can spend a significant amount of their life just idling, and that's not reflected in an odometer reading because the chassis never went anywhere in that time.
I've never replaced an odometer when replacing an engine. What I will do, however, is make a note in the cars documentation that the engine was replaced at xxx,000 miles with a new/refurbished engine with xx,000 miles. I also keep all the receipts of any ancillary parts replaced at the same time (tensioners, water pumps, etc), so that a new buyer can see that the work was done correctly.
This way, you're staying legal, and you're making the replacement engine a positive aspect when you come to sell the car.
You're probably better off not replacing the odometer. You will need to keep accurate track of how far the car has been driven in total, so if you were to later sell your car without disclosing that the actual mileage is the new odometer reading + 220k miles, you could be found guilty of fraud. You certainly don't have to replace the odometer, and it will only increase the amount of bookkeeping you need to do, so it'll be simplest if you don't.