If recalling your dreams from earlier in the night really matters to you, then yes, waking up in the middle of the night will make your brain more aware, allowing you to recall those early-sleep dreams.
Check out this diagram. It shows a typical sleep pattern, complete with the various stages of sleep, including REM, which is typically where you'll find yourself dreaming. For ease of understanding and for the sake of planning your dreaming experiences, equate REM sleep to dream-time.
As you can see, most of your dreaming occurs during the later half of your 8-hour sleep pattern, which means you would have to sleep earlier to dream more during the night, but to recall dreams earlier in your sleep pattern regardless of whether you sleep early, late, during the night or during the day, waking up in the middle of the night will help.
Be aware that this comes at the expense of your restful sleep, though, which is why not all induction methods are suitable for everyone's schedule. If you're still in high school or have a steady job, you might want to stick with recalling your morning dreams and making the most of them, foregoing waking yourself during the night. Weigh your options.

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