Yes Tim, that’s the question which has been persuing human kind forever.
I guess there are multiple answers to that in stead of a single one. First when you have existed you exist in eternity. Death is a relative. There are people remembering previous lives, but I presume this is caused by a 'gene simularity' between the two brain concsiuousnesses. Since eternity is timeless, there can be an instant contact creating the same memory, although this is exeptional. Second, you transfer into other peoples consciousness by sharing, so you don’t even need offspring to continue living. The third option is that your gene consciousness transfers inmediately into a simular vibration, that of a just concepted baby/ but you lose your selfconsciousness.
The fourth (sciencefiction?) option is when it would be possible to transpond a body to another place in time or space, this needs to rebuild your body from the same material/ and would create a double. So you would exist twice :-)