  We extend the microscopic particle-rotor model for hypernuclear low-lying
states by including the derivative and tensor coupling terms in the
point-coupling nucleon-$\Lambda$ particle ($N\Lambda$) interaction. Taking
$^{13}_{~\Lambda}$C as an example, we show that a good overall description for
excitation spectra is achieved with four sets of effective $N\Lambda$
interaction. We find that the $\Lambda$ hyperon binding energy decreases
monotonically with increasing the strengths of the high-order interaction
terms. In particular, the tensor coupling term decreases the energy splitting
between the first $1/2^-$ and $3/2^-$ states and increases the energy splitting
between the first $3/2^+$ and $5/2^+$ states in $^{13}_{~\Lambda}$C.
