FWIW, if you NEED next day delivery, a 1 month Prime subscription is usually cheaper than next day(if available) without Prime. I've done that plenty of times.
Amazon shipping can be a bit of a gamble. Remember that there are basically 3 categories of items sold on the website-items sold and fulfilled by Amazon, items sold by a 3rd party and fulfilled by Amazon, and items simply sold THROUGH Amazon by a 3rd party and shipped/fulfilled independently of them. The first category are items likely to be available next day, while the second category is usually 2 day(minimum) and the last category is is whenever the 3rd party seller gets around to shipping(most are decently fast). It's also worth noting that regardless of your Prime status, most items in the 3rd category do not get any kind of discounted shipping.