How do I tell Matplotlib to create a second (new) plot, then later plot on the old one?

If you find yourself doing things like this regularly it may be worth investigating the object-oriented interface to matplotlib. In your case:

import matplotlib.pyplot as plt
import numpy as np

x = np.arange(5)
y = np.exp(x)
fig1, ax1 = plt.subplots()
ax1.plot(x, y)
ax1.set_title("Axis 1 title")
ax1.set_xlabel("X-label for axis 1")

z = np.sin(x)
fig2, (ax2, ax3) = plt.subplots(nrows=2, ncols=1) # two axes on figure
ax2.plot(x, z)
ax3.plot(x, -z)

w = np.cos(x)
ax1.plot(x, w) # can continue plotting on the first axis

It is a little more verbose but it's much clearer and easier to keep track of, especially with several figures each with multiple subplots.

When you call figure, simply number the plot.

x = arange(5)
y = np.exp(5)
plt.figure(0)
plt.plot(x, y)

z = np.sin(x)
plt.figure(1)
plt.plot(x, z)

w = np.cos(x)
plt.figure(0) # Here's the part I need
plt.plot(x, w)

Edit: Note that you can number the plots however you want (here, starting from 0) but if you don't provide figure with a number at all when you create a new one, the automatic numbering will start at 1 ("Matlab Style" according to the docs).

However, numbering starts at 1, so:

x = arange(5)
y = np.exp(5)
plt.figure(1)
plt.plot(x, y)

z = np.sin(x)
plt.figure(2)
plt.plot(x, z)

w = np.cos(x)
plt.figure(1) # Here's the part I need, but numbering starts at 1!
plt.plot(x, w)

Also, if you have multiple axes on a figure, such as subplots, use the axes(h) command where h is the handle of the desired axes object to focus on that axes.

(don't have comment privileges yet, sorry for new answer!)