  The nano-Hertz gravitational wave background (GWB) is a key probe of
supermassive black hole (SMBH) formation and evolution, if the background
arises predominantly from binary SMBHs. The amplitude of the GWB, which is
typically quantified in terms of the characteristic strain, $A_{\rm 1 yr}$, at
a frequency $1\,{\rm yr}^{-1}$, encodes significant astrophysical information
about the SMBH binary (SMBHB) population, including the mass and redshift
distributions of SMBHBs. Recent results from a number of pulsar timing arrays
have identified a common-spectrum noise process that is consistent with a loud
GWB signal with amplitude $A_{\rm 1 yr}{\sim}2\times10^{-15}$, which is higher
than typical predictions $A_{\rm 1 yr} \lesssim 10^{-15}$. These predictions
usually assume theoretically-motivated but highly uncertain prescriptions for
SMBH seeding and evolution. In this work, we use a simple, flexible model of
SMBH evolution to explore the possible range of GWB amplitudes, given
observational constraints. In particular, we focus on the possible contribution
to the GWB from high redshift ($z\gtrsim 1$) SMBHBs, for which few robust
observational constraints exist. We find that the GWB amplitude may be higher
than fiducial predictions by as much as ${\sim}0.5$ dex if much of the SMBH
mass density was established by $z\sim1$. Beyond pulsar timing constraints,
observations of the high redshift SMBH population from the James Webb Space
Telescope and the Laser Interferometer Space Antenna will be key for
constraining the contribution of high-$z$ SMBHBs to the GWB.
