Another find... Joseph Hitchcock and Jane Swadken had a daughter named Jemima Jane Hitchcock who was baptised 27/Nov/1836 at Saint Mary Magdalene, Bermondsey. Sadly, Jemima Jane died a couple of years later and was buried on 08/Dec/1839 at Saint Mary Magdalene.
It's a long shot but, assuming Joseph was Charles' brother (they were both born in Dedham a few years apart) and Joseph's wife Jane (who would have been pregnant with Jemima Jane at the time) was the witness at the wedding of Charles & Sarah Pike, I'm wondering if Joseph & Jane decided to name their daughter after Charles' first deceased wife (Jemima Spooner)? Possibly because Charles & Jemima had named their last child together Joseph?
I've still not been able to find anything definitive on Jemima Hitchcock née Spooner's death, but presumably she died between 1829 (when Joseph, Alfred Hitchcock's grandfather, was born) and 1836 (when Charles married Sarah Pike). Tantalisingly, there is an entry for what looks like "Jemmiah Hitchcock" (curse hard to read handwriting!) who was buried on 29/Sep/1835 in West Ham/Stratford. The given age is 33, which is a few years out as Jemima Spooner was baptised June 1796. However, I can't link this person back to anyone else in West Ham, so perhaps this actually is the recorded death of Charles' first wife?
The other children of Joseph & Jane were Jane Elizabeth Hitchcock (b. ~1826), Joseph Hitchcock (b. ~1828), Edward Hitchcock (b. ~1830), William John Hitchcock (b. ~1832) and Alfred Charles Hitchcock (b. ~1834).
Finally, if Jemima Spooner was Charles' first wife, then we can likely add Emma Hitchcock to the list of their children. She was baptised 29/Aug/1824 to a Charles & Jemima Hitchcock at St. Margaret, Barking, Essex. Barking is 4 miles east of Stratford and 8 miles east of Bermondsey.