Used gun? Sounds like the chambers need a good cleaning. However, I'm curious about the ejector rod not being long enough. My Blackhawks all have longer barrel, and I assume, longer rods.
You are holding the muzzle up (vertical) when unloading, aren't you? With the muzzle up, the brass should drop free once pushed out (or nearly out) by the ejector rod. If not, a flick of your fingernail against the rim should do it.
You said "often times" but that's not "always" so if this is only happening part of the time, there are several things that could be the reason.
One could be the load being shot. Another is rough chambers, or dirty chambers, or not holding the gun so cases fall free. Too short an ejector rod, or even a misunderstanding of what is "normal".
I'm aware of the problem of the long .357 cases not being full ejected due to the short stroke of the ejector rod in snub nose DA revolvers, never heard of that issue with a Ruger Blackhawk before though.
Clean the chambers. Try some different ammo. Try some .38 Special to see if the shorter brass makes a difference. If it did it all the time, with everything, I'd say call Ruger and talk to them.
Someone might come along and tell you that your need to polish the chambers. Some people will always tell you to modify your gun to "fix" what ever the issue is. I say having work done to the gun is the LAST step to be considered. First steps are to make sure it is a problem, then figure out if its a GUN problem, or an ammo problem, the fit between the two, or a user problem.
One thing the (stock) new model Blackhawk doesn't do is perfectly align the chamber with the loading gate window by itself. I've found that is is possible for a .45 Colt case to rub on the frame window as it is ejected, if the cylinder isn't in just the right spot. I doubt this would happen with the smaller diameter .357 brass though.
I see your situation as more of an irritation, not an actual problem, at this point. There's no "speed drill" for loading/unloading an SA revolver. Muzzle up to empty, down to load. If you have to flick out a fired case once in a while, that's the way it sometimes works. try some different things and see if there is any difference.
Good luck and let us know how it goes!