I am not familiar with all distros (and the boot scripts mechanism is one area where distros differ) but ...
The question is not well phrased because "early in the boot process
" is imprecise. You answer is similarly imprecise because the booting system does not go immediately to the default run level so "the runlevel the system is entering
" begs the question "Which run level".
If "early" is very early, you might want to create the startup link /etc/rc.d/rcS
.d/S01bigd because run level S is entered before the default run level. But few daemons are started in run level S and few in run level <default run level> before S20; as run level S is initialised and in run level <default run level> before S~20, core Linux components are being initialised which are pre-requisites for most daemons. The default for most daemons is /etc/rc.<default run level>/S20; hence a bunch of startup script symlinks are have /etc/rc.<default run level>/S20* symlinks; hence "early" for a normal daemon would be /etc/rc<default run level>.d/~S19. Much depends on whether the daemon requires file systems to be mounted and networking to be fully initialised (initialising networking, including networked file systems, takes relatively a long time).
Regards killing the daemon, this is normally done in /etc/rc..d where the kills are done in reverse order from the startups so a daemon started by /etc/rc<default run level>.d/~S99<something> is killed first with a /etc/rc./K00<something> and a daemon started by /etc/rc<default run level>.d/~S01<something> is killed first with a /etc/rc./K99<something>. This better serves any interdependencies amongst the daemons.