There are two kinds (that I know of) of tiny tags suitable for temporary birth tags/ management tags. These go in at birth (room for up to three numbers of your choice), and stay until the lambs are 4 months old, by which time we've decided which females we want to register. Males are tagged then too, but we wait until they are 16 months before deciding whether to register any, but we don't have to make the tags year dependent.
The turkey wing tags we use as birth tags come from Roxanne of Selkirk. They are tiny metal 'padlock' type things, which are popped in by hand, and they don't leave a hole. Occasionally one comes out, but you can work out who's who easily enough by your method of elimination Rosemary when they're young.
There is another tiny plastic tag, not big enough for all the info the permanent tags require. These fit like many standard tags, but the pin is thin so they don't leave much of a hole. If there is a hole, then the new tag goes into that, without pain, bleeding etc. Someone else may know who supplies these.