After EEA Family permit expiration, how can I stay, work?

Trying to figure out my best option that will allow me to work and stay in UK after EEA Family permit expires. The scenario:

1.) I've been in UK on Tier 4 Student visa (since Oct 2018, expires April 2020). I am trying to land a job & sponsorship to stay in UK, but no luck so far.

2.) Husband is Swiss and will do a TEMP 6 mos fellowship in UK (April-August 2020). I have just applied for EEA Family Permit (from France, our permanent residence) - but will it allow me to work? What if I want to stay in UK and work (after my EEA Permit expires & husband goes back to residence (France)?

Is there any way to extend to a longer residency/ working visa that is not dependent on my husband living in UK? Seems employer sponsorship would be needed...

You will need to qualify on your own (find sponsorship from a company). The moment your husband leaves UK is the moment when your EEA FP ceases, so there is no way he lives in France and you stay in UK as a dependent on him.

A card delivered by France to a family member of an EEA national allows that EEA national to reside and work in France only. That card won't give you similar rights in the UK or any other country.

By 2020, you have a find a Tier 2 sponsor for a work visa. That's usually the dream option but to be fair, I have already seen a few students leaving the UK after their studies without never having found a sponsor. It seems to me like the market is not very favourable to new graduates.

If your husbands comes to the UK and exercises Treaty Right (mind that Brexit!!!), you may get some status under EEA rules during the transition period. However, if he is in the UK on temporary basis, that won't help. Because the time he leaves, you are technically supposed to leave the UK. The minute the Home Office realises that your sponsor is not the in the country, your card is no longer valid.

You can both receive pre-settled status, it will expire in 5 years if visits are made occasionally