There are two components, 'regular' SO2 and HCl and 'unusual' high magnitude H2O (water vapor):
From the NASA interview linked here:
And the first paper that came out about the volcano, they said, no, no, the normal cooling volcanic pollution is still bigger than the warming water vapor component.
From the wikipedia article you didn't link:
One study { of this specif eruption ) estimated a 7% increase in the probability that global warming will exceed 1.5 °C (2.7 °F) in at least one of the next five years, although greenhouse gas emissions and climate policy to mitigate them remain the major determinant of this risk.
Another study estimated that the water vapor will stay in the stratosphere for up to eight years, and influence winter weather in both hemispheres.
More recent studies have indicated that the eruption had a slight cooling effect.
~
https://en.wikipedia.org/wiki/2022_Hunga_Tonga%E2%80%93Hunga...Which is contrary to the time sequence you claim.
FWiW the AGU Letter you did link was an early one (published less than six months after the event, sumitted earlier) and it's inconclusive talking about possibilities such as:
Unlike previous strong eruptions, this event may not cool the surface, but rather it could potentially warm the surface due to the excess water vapor.
'
may not' and '
could potentially'
Either way, according to the NASA interview neither marine fuel change, the El Nino event, nor the eruption combined are sufficient (as modelled, given their error bars) to explain the global increase observed.
According to the NASA interview linked here there are still other factors at play.