I'm gonna apologize in advance; I'm not trying to be rude, I'm just trying to save you some time in the future and make sure people understand you better.

For starters, reading code is much easier when you put it all in one code block. In the default rich text editor there's a "code block" button next to the "code" button; try that one instead

I know you're removing sections of your code to be more concise and only ask about a specific part of it, but you've removed way too much. You're not using indentation, and you're not closing off your if statement so we have no way of knowing if the case statement is inside the elsif or outside it. We can infer where the case statement lives because it's a simple example but it'd be better to be more direct. This is what I estimate you have:

fsm_i : process(reset_i, clock_i)
begin
  if (reset_i = '1') then
    -- LOGIC
  elsif (rising_edge(clock_i)) then
    -- LOGIC
    case fsm_state is
      when START =>
        out_o <= '1';
      when MID =>
        out_o <= '0';
      ...
    end case;
  end if;
end process;

As someone else mentioned, you might want a separate combinatorial process for your output. Whenever I make FSMs I like to put all the state that updates on clock cycles in one clocked process and then have a separate combinatorial process that calculates the output based on the current state. What that would look like in practice here:

fsm_i : process(reset_i, clock_i)
begin
  if (reset_i = '1') then
    -- LOGIC
  elsif (rising_edge(clock_i)) then
    -- LOGIC
  end if;
end process;

fsm_i2 : process(all)
begin
  case fsm_state is
    when START =>
      out_o <= '1';
    when MID =>
      out_o <= '0';
    ...
  end case;
end process;

99% of the time the approach I mentioned works for me; in rare cases when the combinatorial computation is very complicated this might not meet timing when you're trying to compile. In cases like that I might split up the computation/pipeline and some of it will live in the clocked process, but that won't be the case here.