Se você entrar na tela Putty Configuration, na categoria "Terminals", na guia "Features", há uma caixa de seleção para "Desativar alternar para a tela do terminal alternativo". Certifique-se de verificá-lo e você terá exatamente o comportamento que está procurando.
Da documentação do Putty:
4.6.4 Disabling switching to the alternate screen
Many terminals, including PuTTY, support an ‘alternate screen’. This is the same size as the ordinary terminal screen, but separate. Typically a screen-based program such as a text editor might switch the terminal to the alternate screen before starting up. Then at the end of the run, it switches back to the primary screen, and you see the screen contents just as they were before starting the editor.
Some people prefer this not to happen. If you want your editor to run in the same screen as the rest of your terminal activity, you can disable the alternate screen feature completely.