  The Higgs mechanism is a central part of the Standard Model which has not yet
been fully established experimentally without the measurement of the Higgs
self-coupling. Future linear $e^+e^-$ colliders are able to access
centre-of-mass energies of 500 GeV and beyond and can therefore probe the Higgs
self-coupling directly through the measurement of double Higgs production. A
new analysis of the capability to measure the double Higgs-strahlung,
$e^+e^-\to ZHH$, at a centre-of-mass energy of 500 GeV is ongoing based on the
detailed, Geant4-based simulation of the ILD detector concept. This study has
identified several aspects concerning the reconstruction techniques to fully
exploit the detector potential, which are expected to improve precision reach
and will be presented in this contribution. Additionally, the requirements that
the Higgs self-coupling measurement puts on the choice of centre-of-mass energy
will be evaluated as this is important for shaping the landscape of future
colliders such as ILC or $C^3$.
