Which model .44 Special do you have? It's one of the older Smith's, you really should consider going to a .44 Magnum to chunk 300 grain bullets. Going with the heavier bullets really drives the pressures up.
If you are really intenet on doing this, try starting at the min. load for the 240s and work up slowly using a chronograph. You can not rely on what the primers look like to determine pressures. Nor can you rely on whether or not the case falls from the gun. Should anyone tell you differently, I will refer you to any reloading manual of your choice. It's in that often unread portion in the front.
Also, you may want to take a photo of your gun just in case it turns into shrapnel.
A .45 Colt will move 300 grain bullets rather nicely at 1200 fps....something to consider.