By using a slightly different offset you get a slightly different nonlinear transformation, and one that may work even better.

There isn't a way to make this transformation without a choice. You've made a choice by adding $1 -- it looks kind of canonical but really it's based on the size of a dollar, which is pretty arbitrary.

For example say instead of denominating everything in dollars you'd denominated in cents (and added 1 cent before logging). Then everyone would move up the graph by pretty much log(100), except the people who gave nothing, who would be pulled further from the main part of the graph. I think this would make your fit worse.

In a similar way, perhaps you can make the fit better by denominating everyone's donations in hectodollars (h$1 = $100), or equivalently by changing the offset to $100.

We could try to pick the right offset by doing a sensitivity analysis and seeing what gives us the best fit, or by thinking about whether there's a reasonable meaning to attach to the figure. In this case we might think that people tend to give something back to society even when they don't do this explicitly as charity donations, so add on a figure to account for this. My feeling is that $1 is probably smaller than optimal under either interpretation. This would fit with the intuition that going from donating $1 to $9 is likely a smaller deal at a personal level than going from $199 to $999 (counted the same in the current system).