Yes, the 10 watt is plenty to keep the batteries from going dead during storage. This little amount of solar may or may not even have a charge controller. I had 10 (or 15) on the roof pumping in a full 18 volts everytime the sun shown and never ever had battery issues during storage. I had a 5 watt dashboard panel to keep the starting battery topped off and both worked just fine.
These trickle chargers had very thin wires from panel to battery though so I suspect yours does too.
So, no, the answer is, the 10 watt panel does nothing to help you get going on useable camping solar panels.
100 watts would keep batts topped off and let you use lights and fans at night. You really do need 200 watts to truly be self-sufficient out there boondocking. At about 70-80 cents per watt these days off eBay I strongly suggest going 200-350 watts since you are doing all the work, something you might end up with when all is saidanddone anyway.
One 350 watt panel, or two 100 (or 150) watt panels.
You can get two panels and run in series to double your voltage which is more efficient over the wire, just be sure the charge controller can handle the voltage.
18 volts is typical for a "12 volt" panel. So double that and 37 volts would be if you run them in series. Most controllers can handle either automatically.
MPPT is most peoples preferred charge controller. Get the right amperage.