Phil wrote:
"If not, what other homebrew antenna will be fine for this purpose?"

A turnstile with reflector?

High gain comes from narrow beamwidth. A lower gain antenna is easier to
use as it is a larger window on the sky.

Unless the vertical antenna is an axial helix or something similar, it
likely has a null in the direction of its open-circuited end. There are
better choices.

There are many antenna options. Most are covered in the ARRL Antenna
Book. It has a complete chapter on "Antennas hor Space Communications"
In my 19th edition, this is Chapter 19. Construction details are given
too.

Best regards, Richard Harrison, KB5WZI