You could try to use a RACE kit aproach, this will ligate adaptors to single stranded DNA ( for instance with terminal transferase and dCTP.) You could then use a poly-G primer to synthesize the second strand.
Whole genome amplification? Similar to the Klenow, but then with phi29 polymerase instead of Klenow. But do not know if this works (it also makes use of random hexamer primers).
If you have more fragments, different sizes you should synthesized primers for every of them and do the PCR or RACE is a very good solutionor maybe Genome Walking procedure...