There just isn't a lot you can do. It is pretty well understood these days that you aren't leaving your job, but rather you are firing your current management.
If the non-profit agency in question is local, and you are looking locally, everyone you talk to probably already knows it is a hellhole (OK, I exaggerate a bit) and they understand. "Looking for new opportunities", "looking for growth opportunities", are good.
"Looking for increased challenge" may not be good. There are entirely too many outfits out there who see "increased challenge" as synonymous with "bigger pile, smaller shovel, and tighter deadline".
The smartest thing you can do is sit down and think HARD about what you have done, what you enjoyed, what you want to do, where you want to grow, and then start looking for the good outfits who match all that. You MAY have to compromise on some things, but be careful.
Some examples.
I was at General Dynamics/Fort Worth Division (GD/FW), on F-16 project, for a few years. They were lower pay, but you could do things there you just couldn't do anywhere else. You did have to love airplanes. I am told that it wasn't as nice a place after it was sold to Lockheed-Martin.
I was at Texas Instruments Defense Systems and Electronics Group for several years. If it was still around and was still the same outfit I joined, I would most likely have retired from there a year or two ago. (I might well still be there, in harness.) After Jerry Junkins had his heart attack, and the professional managers took over, it went downhill fast and far. I wound up leaving after they were sold, and the new owners "invited" us to move to Tucson AZ. They were really inviting us to leave the company.
Long story. A lot of good people did the same thing.
I was at a small electronic warfare outfit for a year or so. They fundamentally promised high challenge and low BS, and they basically lied on two counts.
At one point, while working at GD/FW, I interviewed a guy. He was not certain how to list his employer, as they'd changed hands a LOT over the last few years. I explained to him that this wasn't a problem: EVERYONE in the DFW Metroplex aerospace and defense community knew the stories, and everyone knew that, despite the frequent sales, they were a very good outfit and they only hired very good people.
I was at Nortel Networks in Richardson TX during their implosion. They were a really good outfit when I started there. Things changed...
Don't worry overmuch about it. Focus on what you want and on finding the right fit.