It is quite usual for experienced woodwind players to 'double' on other wind instruments, and even though the 'blowing' technique may be different I have been led to believe the fingerings are very similar.
But, I would agree with others, Yes, piano, or keyboard, is the ideal second instrument. It gives a more 'graphical' impression of notes, scales etc, and it also goes a long way towards making music theory more accessible. [In due course I hope to produce a 'beginners' guide' to a few essentials of Theory, for interested MNers.]
What 'practical reasons' did you have in mind? Many people are 'snobby' about keyboards and digital pianos, and think only a REAL piano will do, but a huge amount can be learnt from a keyboard that could not be learnt from an acoustic piano.
As a TA in primary schools for over twenty years, I ran a keyboard club for Yr6 children, who brought in their own keyboards, and I encouraged them to make up their own tunes and to start to improvise. I had recorder groups for ten years, from Yr2 up to Yr6, for children who wanted to continue. And I coached children on percussion to accompany the Christmas production, and had a percussion club for a while. All of this was 'informal' music, so children could get a taste of music, and there was no question of 'Grades' as I am not qualified in that area.