Larry,
One of the problems with hunting wit handgun calibers is the "bad news physics" that you encounter.
Great expanding bullets like the Nosler Partition rely on high velocity to generate the force necessary to cause the expansion designed into the bullet. So, the first problem encountered is how to design a bullet that will hold together in flight and at impact but expand at the much lower velocities encountered in handguns...
Ultimately the bullet is at the edge of the envelope in either construction (soft and lightly jacketed) so that expansion is possible (not always probable) or the bullets are made very light (ie. the old SuperVel 115gr) so that velocity can be increased. Either way the compromise brings with it potential problems. And, even the very light high velocity bullets have problems of hit or miss expansion. And, if they are very light, the bullet may expand but may not penetrate deeply enough.
In smaller animals like coyotes there is not a lot of body mass to work with for expansion in any case. So, in smaller animals the wound canal needs to open quickly... and again, at handgun velocities that is difficult to design into any bullet. In a .357 there isn't enough frontal section to generate quick energy transfers so that the wound canal on small animals is generally a "pass through" shot. Probably, like an arrow, ultimately fatal but not instantaneous. So, in trying to find a load that would work in a small animal like a coyote or a deer I would look for something in the 120gr range that is a fully jacketed hollow point. As I recall, Speer/Sierra (one of them) had a couple of very light fully jacketed hollow cavity bullets even as low as 90-100 grs... I would try to work up a 1300-1500 fps load in one of those for coyotes, keep my shots to 50 yards and see what happened (if those bullets are still available). For deer, try the 115-120 gr same style... loaded as hot as possible and hopefullt fired from a 6" for max velocity... Under 50 yards you should still get 1200+ fps and the probability of expansion will go up dramatically. These will still not be "instantaneous" kills but should be effective.