It is useful to look at a detailed map of this area.  For a faster download CLICK HERE (1.7Mb) for a bit sharper rendition CLICK HERE  (4.2 Mb).  This is a 1:250,000 topo sent to me electronically by Joe on which I have overplotted the path of totality.  Just to get you oriented, I have overplotted Northern limit, Centerline, and Southern Limit as thin dark red lines which run diagonally from the SW (lower left) to NE (upper right).  At the SW corner of the map both centerline and Southern limit are below the map "cut off" at 29deg S longitude.

Scroll over to the right side of the map and you will see where the north running road (d) above crosses centerline.  Just for reference I call this location Old Narylco Road Xing and provide refraction corrected local circumstances in the figure.  (I have not applied limb corrections, yet, but they won't effect our site selection).  From here we would get 26 seconds of totality with the Sun 0.5 degrees above the horizon at mid-eclipse.  The azimuth of the sun is 245 degrees (nearly parallel to the path going to the SW).  The road seems to cross centerline here at a relatively flat plain.  From several points around the Old Narylco Road Xing the altitude seems to be about 120 meters AMSL.  So, the question is: Is the view to the SW in the direction of the Narraniappa Hills unobscured?  From this topo the 150 meter contour, which seems to be the high point of the steepest vertical gradient along the lone-of-site to the Sun  is about 5.2 km away.  *IF* the road is at 120 meters AMSL then this point on the ridge is at an angle of 0.33 degrees from the horizontal.  With the Sun only 0.5 degrees above the astronomical horizon this is too close to call from reading this topo!  However, we could follow the track through Old Narylco past "Three Mile Tank" to "Bottom Narranappa Tank" to get to the western side of the Narrahanappa Hills. That location appears to be somewhat elevated and overlooks "numerous small dry lakes" in the direction toward the Sun, which will be 0.7 degrees above the horizon at mid eclipse.  (This seems to be confirmed by a larger scale global digital terrain map, which is explained here). I would suggest this as our "primary site".  Clearly, though with 4WD vehicles we would have flexibility by going overland in regions of dry lake beds if this location is not particularly suitable.

JC: I would not recommend any true cross country (off 4wd track) driving. Summer driving is hard enough on tyres. Small sticks can easily puncture tyres. The summer heat softens the tyre rubber. Stones or sticks deform the tyres enough to  snap the steel belts.  On a work trip into the Gibson desert in March 1993 our two vehicles sustained 14 broken belts on tyres over two weeks. We were carrying 3 spare tyres per vehicle and still ran into problems. The vehicles you rent will only have one spare tyre so you will have to preserve it carefully. Travelling on "small dry lakes" is usually not possible. Most so called small dry lake surfaces are ground water fed soft supersaturated salt crust over wet mud or soft clay.You can walk across what appears to be a hard surface.  A 4wd vehicle will break the salt crust sink and become trapped in the clay and mud. Even with two vehicles,  extraction can be difficult. I'm speaking from experience  I have written a 4wd techniques article in a response to a long series of Q's Michael and Glenn sent me.
 

Tickalara is one of the few places in the path of totality with an airstrip not too far off centerline.  Here is the specification/reference sheet on the Tickalara airstrip.