You don’t.

You probably ran into this:

You’re building values.

So

iex(1)> fun = fn () ->
...(1)>   with a <- 1,
...(1)>        {2,_} <- {3,a} do
...(1)>     :ok
...(1)>   else
...(1)>     {_,a} = error ->
...(1)>       {:error, error, a}
...(1)>     error ->
...(1)>       {:error, error}
...(1)>   end
...(1)> end
#Function<20.127694169/0 in :erl_eval.expr/5>
iex(2)> fun.()
{:error, {3, 1}, 1}
iex(3)>

So you just have to build the value that contains all the information that you need for the error.

The code in the OP would then look something like this:

def some_action(conn, params) do
    with {:ok1, some_result, new_conn} <- some_func(conn, params),
      {{:ok2, another_result},_} <- {another_func(some_result), new_conn}
    do
      new_conn |> redirect(to: some_path(conn, :index)) # new_conn is accessible here
    else
      {:error1, message} -> render(conn, "some_error_template.html")
      {{:error2, message}, new_conn} -> render(new_conn, "other_error_template.html")
    end
 end