Is it possible to create a Hilbert Transform without a symmetric passband?

I have an application where I am not interested in the entire frequency band of a Hilbert Transform so I was hoping to limit the frequency band in order to lower my number of coefficients.

I tried some scenarios in the Filter Designer tool, but this did not give good results.

Is this possible or is there a better way to do this?

Here are a couple of examples:

Working Scenario:

Trying to decrease upper stop band:

I even tried pushing in the lower stop band and this happened:

The full band I am really interested in is [0.01 0.41].

Thank you for any suggestions!