What type of observational constraints are there for reionization? What type of observational evidence is used to constrain reionization? At what redshift?
 A: Types of observations (at least):


*

*Lyman $\alpha$ forest at $z\approx 2.5-6.5$

*Thomson Scattering Optical depth for Cosmic Microwave Background radiation

*Intergalactic Medium temperature  at $z<\approx 6$


Lyman $\alpha$:
One method:
An absorption phenomenon seen in the spectra of background quasi-stellar objects. Based on the neutral hydrogen absorption which changes the quasar spectra and can be used to measure intergalactic neutral hydrogen assuming FLRW metric and homogeneous universe.
Thomson scattering & CMB:
Photon scattering off electrons during reionization introduces secondary anisotropies to the CMB map which can be studied.
IGM temperature:
"Due to low density, IGM cooling time is long and retains some memory of when and how it was last heated, namely, reionized."
This makes it possible to reconstruct, within the limits of some assumptions (e.g. radiative transfer, star formation & chemical enrichment), its thermal history.
A: In addition to the probes you list yourself, Ly$\alpha$ can also be used in another way, namely to look at the luminosity function (LF) of Ly$\alpha$-emitting galaxies (LAEs), i.e. the number of LAEs at a given luminosity. Since a more neutral IGM scatters Ly$\alpha$, the amplitude of the Ly$\alpha$ LF decreases toward higher redshifts.
Since large galaxies are able to ionize a bubble in the IGM around themselves where Ly$\alpha$ is more easily transmitted, they may make themselves relatively more visible at higher redshifts than smaller galaxies. Hence the slope of the LF can also be affected.
See e.g. Dayal et al. (2010), Tilvi et al. (2010), and Dressler et al. (2015) and references in these papers.
