When should I get news of a 3 year extension?

[hizoto]

Currently under a H1B with expiration date this October (first 3 years). Since we are almost past April, I understand that my employer has now a window of 6 months to request an extension of my visa for another 3 years. Correct? When should I begin to check with them that this will be done? (I am worried that they will take this opportunity to let me go because of recession)

Also, can I switch jobs before the extension is filed and then ask my new employer to request the extension?

Thanks in advance for any response.