  Recently, the so-called ``consistent thermodynamics'' of the Lorentzian
Reissner-Nordstr\"{o}m (RN)-NUT-AdS$_4$ spacetimes has been pursued by a lot of
efforts via different means. Among these attempts, we had proposed a novel idea
that ``The NUT charge is a thermodynamical multihair" to successfully tackle
with the subject. In this paper, we will adopt this strategy to reconsider the
mass formulas of the RN-NUT-AdS$_4$ solutions but written in an alternative
form, which had not been studied before in any existing literature and might be
a most appropriate ansatz for the higher dimensional multiply NUTty-charged AdS
spacetimes without any constraint condition. Here, we shall discuss the
Christodoulou-Ruffini-like squared mass formula and the first law as well as
the Bekenstein-Smarr mass formula by introducing the secondary hair $J_n = Mn$.
For the sake of generality, we have introduced a dimensionless constant $w$
into the constant factor $\Xi$ in the solution expression so that when $w = 1$,
all obtained results can reproduce those delivered in our previous work.
