Are there methods for iron content determination in spinach that do not require thiocyanate?
There are iron testing kits in water which are available at an affordable price. I have not used them but read more about them here Testing iron in well water. You can test your spinach extracts for a yes no type answer if iron is present in significant quantities or not. Just keep in mind that PopEye's spinach is really not that rich in iron. Too much iron is a liver toxin.
I am afraid to share the bad news for you that with the resources you have, you cannot reliably determine iron content in spinach. Micronutrients or minerals cannot be determined by classical methods because they are present in a complex matrix and in very small quantities (parts per million range). They need instrumental analysis (absorption by molecular complexes, atomic emission or atomic absorption). I saw a couple of papers from ResearchGate which talk about the titration of spinach extract with permanganate. Nothing can be more wrong than this procedure in analytical chemistry. They are basically titrating all the organic stuff with it. Those methods are horribly wrong. Someone on SE Chemistry posted a query on how to determine iron in spinach by titration Iron in spinach...that is all incorrect. Yes, you will generate some numbers with those burette readings but those numbers have no significance.
Regarding your ashing method, that will not work either, because you need at several kgs of spinach to get a small amount (milligram) of iron. What should be the ashing temperature? How would you isolate iron from the ashes? Selective precipitation to get rid of other minerals? Nope, that is a bigger headache. For that you would need more reagents.
If this is a school project, make sure you have the resources after searching the literature. If this is an home experiment- classical methods will not work.