Is it possible to further fragment an IP fragment?

Solution 1:

It is possible to further fragment IP fragments at any point on the path.
However, the 're-assembly' is done only at the destination -- so no 'de-fragmentation'.

Reassembly at the receiving end-host is done based on the fragment-offset and more-fragments flag in each fragmented IP packet header. The Identification field help collect all the fragments of an IP packet together.

^{You might also be interested to know the host of problems this introduces.
And, IPv6 handles packet-life differently.
Another interesting ref: Resolve IP Fragmentation, MTU, MSS, and PMTUD Issues with GRE and IPSEC}

Solution 2:

Yes, this is possible. Reassembly is possible because each fragment contains an offset which is relative to the original payload, not the fragemented payload. You can read more here http://penguin.dcs.bbk.ac.uk/academic/networks/network-layer/fragmentation/index.php.