Hi, more knowledgeable people than me will be along but here goes my comments.
The ammunition you show is probably collectible, so don't shoot it. Since it is a lead bullet load, regardless of the claims that might be on the box it is not true "metal piercing" ammunition. But it will shoot through thin metal.
The original .38-44 ammunition was loaded to higher velocities, approximately 1150 fps with a 158 grain bullet, over the standard .38 Special. It was advertised as a more powerful round to penetrate the automobile bodies of the 1930's when it was introduced. As such it is considered too "hot" pressure wise for standard size .38 Special revolvers. This means that the round could be safely fired in the S&W "N" frame revolvers, the Colt New Service revolvers, and rarely spoken off but could also be fired in the Colt Official Police revolvers.
It was introduced in 1930?? and was exceeded by the introduction of the .357 Magnum in 1935. However the S&W revolvers that were made for the round, the Heavy Duty and the Outdoorsman, were made until 1960??. And in large numbers considering the production of the .357 revolvers.
And be forewarned, if you do not post pictures of your Outdoorsman many people here will claim you do not own one..........in other words we would love to see pics of your hogleg.