I did this to clone an alternatively spliced cassette into a gene. See my supplemental methods in 2015 Gonzalez et al. "Tight regulation of plant immune responses by combining promoter and suicide exon elements". https://academic.oup.com/nar/article/43/14/7152/2902986
Main takeaway: your PCR pieces need to overlap enough to act like internal primers. For us, ~25-28 bases over overlap was enough. Less is fine if you have a higher GC ratio for your middle piece (our middle piece was high AT). Use a Tm calculator like IDT's Oligo Analyzer to measure the melting temperature of ONLY the sections that overlap. The Tm should be high enough to work in a normal PCR.
Take your A260 nanodrop reading and PCR product sequences to measure the molar concentration of the DNA. There are online calculators that can help. Don't use ng/ul concentrations because they won't be accurate enough.
Set up a PCR with a high fidelity polymerase and use your PCR pieces as templates. When I did it, I used 1:1:1 equimolar concentrations of my three PCR products, Phusion DNA polymerase, and the normal forward and reverse primers for the gene getting the splicing cassette (avrBs2). Run your product on an agarose gel and cut out the size of your stitched-together desired product. I suggest setting up a few slightly different reactions at the same time in case you need to troubleshoot this. Try different molar ratios, different PCR buffers (or PCR additives like DMSO), and different PCR temperatures.
Good luck. Remember to sequence verify your final product!
Just to clarify, I’m trying to ligate a mixed population of PCR fragments into big fragments which I’ll use for long read NGS. I’m considering using complementary primers to form a linker with a T overhang at each end, and mixing it with the PCR products in a ligation reaction. Hopefully that’ll give large multimeric fragments of the PCR.
However, I'd like to know if there’s a standard way to do it that would be better/easier for my students.