Either one of the expander plugs pictured above will work with a 310 tool. Lyman seems to have difficulty figuring out just what these kinds of gizmos should look like (see pic below). On top they will be marked either .357 or .358. I have both.
I have put together a rather complete set of 310 dies for the 38 Special/357 Mag, will several expanders and seating dies with plugs. I use an pre-war Lyman/Ideal No. 3 set of handles. I load lots of ammo with it.
I also have a Lyman Tru-Line Jr. press set up just for 38 Special. It has a FL size die which BTW sizes 357 Mag cases as well. Lyman can be tricky about such things, stamping the same die 38 Spl. or 357Mag to sell more dies.
Lots of fun to be hand with old school reloading tools and the ammo produced is as good as what can be produced on a $2,000.00 super duper double thrown down progressive press.