Like Jim says, Smith Corona was one of the contracted makers of the 1903A3 Springfield rifle. They were made in .30-06 only. Your rifle very likely is one of them that has been "sporterized". This isn't unusual, but it means there is no collector value. Not that value matters for a rifle you inherited. Inheriting any rifle isn't a bad thing. Inheriting an Smith Corona '03A3, even a sporterized one, is friggin' great.
Does the forestock end about half way along the barrel?
Are there any markings on the barrel itself? You may have to take off the stock to find them. If there are, post all of them.
If there are no markings, it has an aftermarket barrel that may or may not be a .30-06. No markings will also mean you should go to a smithy to have a chamber cast done and the barrel slugged(you could do that yourself. Hammer a cast .30 calibre bullet through the barrel using a brass rod and a plastic hammer and measure the bullet with a micrometer. A .30-06 bullet will measure .308".) Look for any markings and post them first though. There are lots of .30 calibre cartridges.
Post the serial number without the last two digits, (put 2 X's) and somebody will give you the year and month your rifle's receiver was made. It'll be sometime between 24 Oct., 1942 and 19 Feb., 1944. SC made a total of 234,580 rifles. SC was the only W.W. II manufacturer of the '03A3 rifle other than Remington. Full military configuration(no scopes) SC's are highly prized by collectors and shooters.