  We search for possible pulsar TeV halos among the very-high-energy (VHE)
sources reported in different VHE surveys, among which in particular we use the
results from the first Large High Altitude Air Shower Observatory (LHAASO)
catalog of $\gamma$-ray sources. Six candidates are found. They share similar
properties of containing a middle-aged, gamma-ray--bright pulsar in their
positional error circles (the respective pulsars are J0248+6021, J0359+5414,
J0622+3749, J0633+0632, J2006+3102, and J2238+5903), being in a rather clean
field without any common Galactic VHE-emitting supernova remnants or (bright)
pulsar wind nebulae (PWNe), and showing an absence of any gamma-ray emissions
in 0.1--500\,GeV after removing the pulsars' emissions. Combining these
candidates with several reported (candidate) TeV halos, we obtain the
relationships between their luminosity at 50\,TeV ($L_{\rm 50TeV}$) and the
corresponding pulsars' spin-down energy ($\dot{E}$), which are $L_{\rm
50TeV}\sim \dot{E}^{0.9}$ and $L_{\rm 50TeV}/\dot{E}\sim 6.4\times 10^{-4}$.
The relationships are nearly identical to previously reported ones. We probe
possible connections between the extension sizes of the VHE sources and the
pulsars' ages, and find a weak older-and-smaller trend. By comparing to the VHE
detection results for PWNe, it is clear that the (candidate) TeV halos have
hard emissions by either having their power-law indices be smaller than 2 in
1--25\,TeV or by only being detected in 25--100\,TeV. In addition, we also
consider seven other VHE sources as possible TeV halos based on the results
from different studies of them, but they do not fit cleanly with the properties
listed above, indicating their potentially complex nature.
