Multiplying by 1000 would only change the scale parameter (of either the gamma or lognormal); if it didn't fit before you multiplied by 1000 it won't fit any better after. So that's not help at all.
If you have exact zeros, no continuous distribution on the positive half-line will really address that problem.
I'd suggest considering a zero-inflated distribution (essentially a mixture of exact 0's and whatever continuous distribution is reasonable for the rest of it). Zero-inflated gammas seem to crop up reasonably often.
You may also have an issue if those percentages can't exceed 100%, because neither the gamma nor the lognormal are bounded like that; in that case you might consider a zero-inflated beta. If 100% is possible, there's also a 0/1-inflated beta.
As for how to model "near-zeros" that entirely depends on what those near-zeros behave like. You haven't given any information on which to base a suggestion of a model -- we'd only be guessing. It may be that an additional mixture component for very small values might help.