Use the break-pane and join-pane commands. Refer to man tmux for details, options and usage.
Hide Pane 3:
Select pane 3 and enter Prefix-:break-pane -dP.
tmux will send pane 3 to a window in the background (the -d flag) and print some information about it in pane 2 (the -P flag). By default you'll see something like 1:2.0 (meaning: session:window.pane). Hit q to continue working.1
1With some practice you you will be able to drop the -P flag since you can predict the session:window.pane triplet: session defaults to the current session and pane defaults to 0 while window will be the next free window identifier.
Get Pane 3 back:
To get pane 3 and the layout back, select pane 2 and issue Prefix-:join-pane -vs 1:2.0 telling tmux to split pane 2 vertically (-v) and to join the (source) pane (-s) with identifier 1:2.0. Optionally you can drop either the session or the pane identifier. Note also that tmux stores a command line history, conveniently accessible with Prefix-:-Up.
You'll probably need some time to get the hang of it, but once you do, you'll surely be able to come up with custom key bindings that are convenient for you.
The question Moving tmux pane to window contains some useful information and tricks that might improve your workflow.
- join-pane [-bdhv] [-l size | -p percentage] [-s src-pane] [-t dst-pane]
(alias: joinp)
Like split-window, but instead of splitting dst-pane and creating a new pane, split it and move src-pane into the space. This can be used to reverse break-pane. The -b option causes src-pane to be joined to left of or above dst-pane.