  In a previous paper, I demonstrated the accuracy of simple, precessing, power
ellipse (p-ellipse) approximations to orbits of low-to-moderate eccentricity in
power-law potentials. Here I explore several extensions of these approximations
to improve accuracy, especially for nearly radial orbits. 1) It is found that
moderately improved orbital fits can be achieved with higher order perturbation
expansions (in eccentricity), with the addition of `harmonic' terms to the
solution. 2) Alternately, a matching of the extreme radial excursions of an
orbit can be imposed, and a more accurate estimate of the eccentricity
parameter is obtained. However, the error in the precession frequency is
usually increased. 3) A correction function of small magnitude corrects the
frequency problem. With this correction, even first order approximations yield
excellent fits at quite high eccentricity over a range of potential indices
that includes flat and falling rotation curve cases. 4) Adding a first harmonic
term to fit the breadth of the orbital loops, and determining the fundamental
and harmonic coefficients by matching to three orbital positions further
improves the fit. With a couple of additional small corrections one obtains
excellent fits to orbits with radial ranges of more than a thousand for some
potentials.
  These simple corrections to the basic p-ellipse are basically in the form of
several successive approximations, and can provide high accuracy. They suggest
new results including that the apsidal precession rate scales approximately as
$log(1-e)$ at very high eccentricities $e$. New insights are also provided on
the occurrence of periodic orbits in various potentials, especially at high
eccentricity.
