Returning to the UK as a tourist after tier 4?

[fungibleone]

Hi all. I'm in a bit of a difficult spot and I'm hoping to get some advice.

I've been in the UK for 4.5 years on a tier 4 visa (for my bachelor's and master's degrees). I'm from the US and got the 6-month extension for my master's degree, but my degree took longer than anticipated so I'm left with less than 2 months until my visa expires and I'm feeling very lost.

I realise the obvious solution is to find a job that would fulfil the requirements for a tier 2 visa, but with so little time left I'm really not confident that this will work. I'm still applying for jobs and hoping for the best, but I know it's not something I can count on.

I would also like to do a PhD and am working on putting together proposals. I'm going to apply for an April start, which should be ok visa-wise, but it's cutting it pretty close and the university may ask me to defer to July or October instead. If this happens, I'll end up with a few months between tier 4 visas. What can I do in this case? I can't return to the US, I have nowhere to go there. Would it be possible to exit the UK and return under a tourist visa? I've read a few horror stories about people being denied entry for this, but I'm wondering how likely that is to happen, especially now that US citizens are allowed to use the e-gates at most UK airports. Would that make a difference at all? Would coming in through a smaller airport (as opposed to Heathrow) be any better or worse?

I'm just feeling so very lost at the moment. I had intended to use this 6 month period to apply for jobs, but having to use 4 of those months to finish my master's was unexpected and has really complicated things. I'm assuming there's no way to extend my current tier 4 visa. I'm just stuck with few ideas left. Any advice at all would be greatly appreciated. Thank you in advance.