Not clear from the question. Is this about service or branch circuits?
Since you upgraded the service, you need larger feed wires to match. Instead of 2 sections of wire and a splice, get wires to go the whole distance.
If you are simply putting in a bigger panel (based on a previous question, I think you really do have upgraded service) then:
- you can splice inside the old panel
- you need to use appropriately sized connectors, which won't be wire nuts at that size
It is quite common to reuse the old panel as a giant junction box. You can splice each connection inside the box with wire nuts. However, keep in mind that if you use conduit > 24" (and question says 15 feet), you have limits as to how many circuits per conduit. If you use individual cables, that's fine - but you can't just bundle them all together for the 15' run - they need to spaced out.