How to move the y axis scale factor to the position next to the y axis label?

You may set the offset to invisible, such that it does not appear in its original position.

ax.yaxis.offsetText.set_visible(False)

You may then get the offset from the formatter an update the label with it

offset = ax.yaxis.get_major_formatter().get_offset()
ax.yaxis.set_label_text("original label" + " " + offset)

such that it appears inside the label.

The following automates this using a class with a callback, such that if the offset changes, it will be updated in the label.

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.ticker as mticker

class Labeloffset():
    def __init__(self,  ax, label="", axis="y"):
        self.axis = {"y":ax.yaxis, "x":ax.xaxis}[axis]
        self.label=label
        ax.callbacks.connect(axis+'lim_changed', self.update)
        ax.figure.canvas.draw()
        self.update(None)

    def update(self, lim):
        fmt = self.axis.get_major_formatter()
        self.axis.offsetText.set_visible(False)
        self.axis.set_label_text(self.label + " "+ fmt.get_offset() )


x = np.arange(5)
y = np.exp(x)*1e-6

fig, ax = plt.subplots()
ax.plot(x,y, marker="d")

formatter = mticker.ScalarFormatter(useMathText=True)
formatter.set_powerlimits((-3,2))
ax.yaxis.set_major_formatter(formatter)

lo = Labeloffset(ax, label="my label", axis="y")


plt.show()

enter image description here

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