Spire contamination missions - and anything else you can do at a spire site - add a flat progress boost to the outer ten systems of a maelstrom, whether they're in Alert, Invasion or Control state.
This boost caps out at 85% progress so needs to be finished off with conventional attacks unless an outermost system contains a spire site itself - then, this progress can go all the way to 100% in that system, which is the only way to recapture a Control system containing a spire. It doesn't matter in that case whether the attacks are on the outermost spire or a separate inner one - the progress is the same either way.