Now that the event is over, the sixth coil is a carousel with 5 possible rewards at the end: one option is always available, and each burden unlocks one. <a rel="nofollow" class="external free" href="https://fallenlondon.wiki/wiki/Exiting_the_Sixth_Coil">https://fallenlondon.wiki/wiki/Exiting_the_Sixth_Coil</a>

So, there is no point in picking up more than one burden. Also, if you want a burden, you have to pick it up going in (ie before you "resolve to leave"). Maximum Patrolling the Labyrinth (for most items from rewards) is 40, so ideally you should turn around ("resolve to leave") when Patrolling the Labyrinth + Coiling Ever Deeper = 40 so as not to waste extra moves.