How to log in where I left off
Unfortunately, there is no way to log back in to where you previously were. Personally I find this quite useful, especially before you find a magic mirror, as sometimes it can be difficult to get back to spawn or your base.
Now one thing that I have done in the past to resume what I was doing previously, or to change where I respawn for a boss fight, is to make a rudimentary room with a bed in, so that I can respawn where ever I would like to.
Although it may be a pain to have to make a room every time that you want to leave the game when you want to preserve your location, this solution may be just the thing that you are after. For information regarding constructing a proper room, please refer to the official Terraria wiki.