I'd look at what users are used to seeing. On mobile my experience is with Android, and it's pretty consistent that menues on the left can be viewed with the following icon, known as the "hamburger icon":
Placement of this is also important. It's usually in the top left of the screen as part of a header bar. Provided it's near the top left I think a user would find it easy.
The problem with swiping, is unless it's something common between applications (like swipe to remove from list) it's not something with any visual cues as a possible action for the user. They don't know they can do it. To show them, you have to spend time developing "tutorial/did you know you can" modes that occasionally start. I don't know about you, but I've always got annoyed with those butting in when I'm trying to use an app. It's never the right time, and even if I did follow it I'll probably not remember.
It's not very common to see a narrower icon menu, but to simply hide the menu completely. Does that mean you shouldn't do it? Of course not. But this is the kind of layout I'd expect the user to be familiar with:
(Taken from Google's Material Design which is a popular mobile design pattern.)
So I'd consider trying to bring a little of this, at least the button placement. You could have it overlapping the header image (but does that scroll off the screen?).
I'd be careful just having it as the first item in the list, as it's almost always in the very top left of the screen and your menu starts a way down the page.
So I'd consider something like these two (excuse the bad Paint editing):
But as the other answer mentioned, if your menu has as little as 3 or 4 items in it and they are consistent over the whole application, why not consider bottom navataion, or tabs? Horizontal space is at a premium with portrait mobile applications. The hamburger menu is somewhat disfavoured these days as it hides things from the user: