Ok, I feel like a bit of a dope asking a rules question for a game that is very simple. But ...
After the first turn, can you color in a bulb not connected to your string?
I assume so, because the rules give two turn options, and do not say that the first option (color in a bulb not part of a string of lights) is only available on the first turn.
On the other hand, the only image of this game is labeled "game end" but there is an unclaimed bulb. Why would the last player not have claimed this bulb if it were possible to claim bulbs unconnected to light strings? Also, the game end condition is when there are no more bulbs to be claimed, so this image further confuses me.
On any turn you may jump to any unclained bulb. I put a note on the end game picture which was obviously played by other rules since players might have verry different number of connection but shouldn't ever have more than a 1 bulb lead.