Hi Kelly,
Sorry for the late answer here.
Most of the time, my web maps will contain a variety of layer types such as: feature services, map services, hosted feature services, tile layers (map or vector), ...
And the rule is that feature services must always sit on top of map services. There is always a way to work around this by playiong with the way the data is published (AGOL as a feature service or a map tile, or directly onto a GIS server).
I have also tested your suggestion of moving a layer to the basemap - it can be a useful solution for some situations.
But the capability of being able to organize layers in any possible order desppite their type (feature or image layer) would save some handeling time.
I will be looking forward to seeing this happening in a near future, as it may simplify some of my workflows.
Alix