EDIT: updated to account for potential $log(0)$ error
You can convert the answer into log space. If you take the output and then just take the log it will radically reduce the difference.
If before your regressor outputs some number from input data $x$:
$$y = f(x)$$
Then you can just take the log of the output with a constant added to avoid $log(0)$ error:
$$y = log(f(x)+c)-c$$
Alternatively, you could just "bin" the output and make it a classification problem- perhaps a new class for every $0.1$ range. This will make the problem much easier to train, but of course is only doable if you're ok losing that precision. Combining log space with classification are also not mutually exclusive.