The Earth's initial accretion was about 4.5 billion years ago, and there is good Hf-W isotopic evidence that an iron core started to form within about 10 M years, and may have been largely complete within 30 M years. However, the Earth's dynamo, which is driven by isotopic heating and core rotation/convection, didn't switch on strait away. It must have built up over hundreds of millions of years, possibly kick-started by a the magnetic field of a stronger solar wind at that time. The evidence from 3.5 Bn year old dacites suggest that the magnetic field at that time was only 30 to 50% of the current value. There is no magnetic data for 4.4 to 3.5 Bn years, (the time period you are interested in), but current models lean towards lower rather than higher values. There appears to be no mechanism for strong magnetism in the early Earth. Geomagnetic evidence from about 2.5 Bn years ago seems to indicate that the Earths magnetism was more stable then than now, with few if any peaks of high or low magnetism. Probably the Earth's early magnetic field will always be imprecisely known because nearly all the early rocks have been 'cooked' in such a way as to extinguish the early magnetic evidence.
