Are objects that fall into a black hole *always* silhoutted against the Hawking radiation coming out?

There are many questions on this site about whether you can really fall into a black hole – some seem to say clearly yes, some say clearly no. So let’s try one more stab at this one with a narrower question:

If you drop a large number of sodium ions into a black hole, and you live a really long time and from many angles dutifully watch the emissions from the sodium ions and the absorption of the Hawking radiation by the sodium ions once it gets bright enough near the end, is there ever any time (right up until the evaporation of the hole) when the sodium ions are not in front of (at a greater radius than the source of) the Hawking radiation?

Some presumptions here: (a) you never stop receiving photons from anything dropped in a hole, and (b) the Hawking radiation originates right at the event horizon. Please feel free to dispute these.