The ferrite material - it depends on the bands you target - for 80-10m the material 43 is an optimal choice, for higher bands like 10-4m I would use material 61.
How to check you ferrite material - made 2-3 turns over it and do measure the inductance, calculate the AL, material 43 is around 900nH/turn2, material 61 is around 100nH/turn2. There are other parameters to consider as well, but most probably the deflection ferrites are not intended for HF, imho.
The ferrite rods from old radios are also not suitable, as those work till couple of MHz only. There are companies selling big ferrite rods for such antennas, like 10cm diameter and 1m long (heavy, expensive), search for it..
PS: You can make S11 measurement (with 1-2 turns) from freq1 to freq2 (I did with nanovna) and make difficult calcs in matlab to get mu' and mu'' graphs which do represent inductance and core losses vs frequency and are usually depicted in the datasheets.