How to fight Omega Flowey a second time

From the wikia page on Flowey:

If the protagonist loads or finishes another Neutral Route, Flowey does not absorb the SOULs because he knows that they would rebel again. Instead, the protagonist will be able to exit the room and receive Sans's Neutral Route phone call.

Due to the part I bolded, you will not be able to fight Flowey in another Neutral Route playthrough without doing the following, based on this thread.

Date Papyrus and befriend Undyne. Then Undyne will give you a letter. Give it to Alphys and go to her lab to find a new level. After completing it you'll get a new ending. After that ending, you can do a True Reset to fight Flowey again.

Thus, to fight Flowey again in a Neutral playthrough, you will need to perform a True Reset, which apparently can only be done after completing a True Pacifist playthrough. (See here)

A True Reset is a function available after completing the True Pacifist Route. It thoroughly erases all of the information stored in the SAVE file, with one exception: If the Genocide Route is completed, the First Human destroys the world and its timelines. They then offer the player to recreate the world in exchange for their SOUL. Accepting this offer is similar to a True Reset; however, accepting this offer also activates a permanent flag in the game's files, which affects the outcome of the True Pacifist Route.