The rule states that we should inclued the Spooktober Jam logo as either a splash screen or on the menu cover. However, it doesn’t specify when the logo should be put. Can I only display the jam’s logo as a splash screen after the end or I should do it once the game begins?

Your choice on when exactly, though the splash screen and main menu are the easiest for us to see right away.  If you want to display the logo after the player hits "Play", that's fine. It's not a very strict rule, just something we ask to help judges identify games as being part of the jam, and help players discover other entries from the jam.