Maybe your bloodshot eyes are caused by lack of sleep in the night. You can go to bed earlier next time. Usually we need to sleep for 7-8hours in the night. But if you sleep too later, it will cause bloodshot eyes too. I will go to bed at nearly 10:30 o'clock every night. I never have bloodshot eyes in the morning except my eyes are infected.
Maybe you are allergic to something in your bed especially on your pillow. You can wash all your stuffs in the bed. If this doesn't work, you can see an eye doctor. Do you drink every night? If you are a heavy drinker, alcohol can lead to bloodshot eyes. You must stop drinking or drink less.