Not sure. Looks like you should be able to recork with a real cork. Just be carefull you use the number 8 and not the reguler number 9's.
A freind guifted my mother with two bottles of homemade cordials. The bottles look like your bottles, but with a flat bottom. They had the tapered corks as well. They are not empty yet and I only have number 9 corks. Otherwise I would try to cork one for you!