I don't think so without doing it explicitly yourself.
I tried tracing the ORDS sessions and also dumping out context variables, and got nothing of value pertaining the call.
You could edit the REST definitions to set a context for the duration of the call, eg
Before
begin
update emp set ename = :ename, job = :job, hiredate = :hiredate
where empno = :id;
:status := 200;
:location := :id;
exception
when others then
:status := 400;
end;
After
begin
dbms_session.set_context('MYCTX','RESTPT','/hr/ermployees/:id/PUT');
update emp set ename = :ename, job = :job, hiredate = :hiredate
where empno = :id;
:status := 200;
:location := :id;
exception
when others then
:status := 400;
end;