Re enter just before the visa ends get your 15 months. Then apply 60 days extention. Gives you a year and five months. Got to do a border bounce 4 times
Leave and re-enter on the OA before it expires and get stamped in for another year, you'll need the insurance cover. Failing that you could do as you ask and re enter visa exempt and apply for the Non Imm O in country if you meet the financial requirements.
Short haul you might get away with it. I dont know any long haul airlines letting you board without return/onward travel unless you have a long term visa/extention of stay or re-entry permit.