A lot of factory .44 Spl. hollowpoint rounds are loaded kind of light. Don't get me wrong, I think some of 'em are still fine choices, but you have to remember that there are still a lot of flimsy Charter Bulldogs out there, and any given .44 Special load is likely to be put in one of them.
My favorite from the big ammo companies is CCI Blazer 200 gr JHP .44 Special. It uses the excellent 200 grain .44 "Special-class" Gold Dot bullet -- a wide mouth hollowpoint for good expansion even at medium speeds. Fairly easy to find in stores.
Now, you can get the identical bullet in the Speer Gold Dot .44 Special factory load, which uses nice reloadable nickel-plated cases instead of the Blazer's disposable aluminum case. But I still think the Blazer version is the better buy (especially since you say you do not reload). It is slightly hotter than the Speer nickel-case version; same bullet; just as reliable and accurate; and it's definitely cheaper. You can get the CCI Blazer for around $21 / 50 ; not peanuts, but the Speer version is darn near $1 per round!
Also from a big maker, there's the Winchester Silvertip 200 gr .44 Special. It's quite lightly loaded -- probably around 750 fps from your gun. More of a Charter Bulldog load. But it reportedly expands well even at those mild speeds. Accurate round. I'd stick with the Blazers - cheaper and more powerful.
Federal makes a lead hollowpoint .44 Special defense round, but I rarely see it and have no experience with it.
Since you are choosing SD loads for a .44 Magnum-rated revolver, you should also think about the Speer 200 gr Gold Dot "Short Barrel" .44 Magnum defense round. It appears to use the same excellent Gold Dot as the two CCI/Speer .44 Special loads listed above, but revved up to around 1050 fps. That's a great power range -- serious thump but much more controllable than full-boat .44 Magnum rounds. I have tried a box of this "Short Barrel" .44 Mag Gold Dot in my 4" S&W 629, a revolver quite similar to the Ruger you're looking at. I truly would not want to go up against a practiced home defender who had a good DA revolver full of those loads. Mean stuff. Expensive but worth it.
Among smaller ammo companies, Buffalo Bore makes a couple of varieties of "hot" .44 Special ammo, not approved for use in Charter guns, that would be great in a Redhawk. One uses a lighter-weight JHP at high speeds. The other is their "Keith-style" load: a 255 gr wide-meplat SWC at about 1000 fps. Great outdoors load. Again, expensive but top quality.
There are probably some other choices from small companies like Double Tap or Georgia Arms with which I'm less familiar. Perhaps others will share info about those.