short magnums, 2-3/4 " magnums
Let me start by saying that I do the same thing now and again, ....carry my Grandad's old pump gun, the one I took my first gobbler with in 1980, on a spring hunt. Fixed full choke, 28" bbl., solid rib, all steel and walnut, Savage Model of 1921! The proverbial blast from the past. But note that I have been advised by competent authority not to shoot the ammo I am about to describe in Pap's old gun. But that is another story.
What is missing from the OP on his "grandpa's old gun" is choke information. I am going to make the assumption that the old gun is fixed "full" choke. I cannot in good faith recommend that any untested modified, or any other more open choke constriction go gobbler hunting. There are some "modified" barrels that shoot considerably tighter than one might think, especially with modern ammo and some testing would yield if one's modified gun would be acceptable. But I hesitate to give a go ahead for every modified gun.
Also, some grandad's are younger than others, and it may be that the gun in question has choke tubes, if so, than the OP needs to acquire, minimally, a standard full tube, and optimally, a special turkey choke.
All that said, there are indeed "short magnums, or baby magnums", 2-3/4" mags that typically throw 1-1/2 oz of lead shot, a quarter oz more than standard heavy field loads. I have in my assortment some old 2-3/4" mags that throw 1-5/8 oz of shot, very well I might add, from my 3" guns. All (well the big 3 anyhow) the domestic ammo makers offer baby mags or short mags, some searching should yield results.
When bamaboy was 16, we used some of the vintage 1-5/8 oz (#6 lead) loads for him to take his first two gobblers...cleanly, at 32 and 46 paces, respectively. The short mags kicked a tad less than 3" shells, so I thought. and him being a bit lanky, the short mags seemed a good choice.
Let me add that it is not unusual for any shotgun not to deliver its payload precisely at the point of aim, again, some simple test shots will tell. If that is the case with the "old gun", then some type of adjustable sights, or Kentucky windage will be called for.
Don't move......he's coming!!!!!!!!!!!!