What do I do, however, for those frequencies for which it is close
to a half wave? PI network? Or L network?

===================================

You very slightly de-tune (decrease) the antenna length, not enough to have
any effect on its radiating properties, such that a simple parallel tuned
matching circuit will cope.

You can connect a coax feedline line either in series with the coil, an
L-network. or tap the coax a few turns up the coil. A very modest ground
connection adjacent to the bottom end of the coil witll suffice.

The same circuit will cater efficiently for quite a wide range of antenna
heights.

I hope you are satisfied with the above description.
---
Reg, G4FGQ