your sample may have short, medium and long range order. and have more than one ore two phases. I had numbered some features in your attached file. 1- white halo confirm the presence of ratio of amorphous phase of the material. 2 and 3 are diffused rings confirm the transformation to some degree of poly crystallization phase 4- white spots confirm the presence of high crystalline phases. so you need for XRD to confirm and titled these phases. For more details i had attached another XRD, SEM and TEM micro graph and one paper to simplify this discussion.
I agree, SAED shows bright reflections from single-crystal, while there are also weak and diffused rings visible ( from other material, since the rings and diff spots does not coincide) - the rings may originate from some oxide layer or other alteration of the sample, or may be an artefact from support material from the TEM grid? We don't have much information about the sample, preparation, treatment etc, so also the answers will be, well, quite general.
what I see on your diffraction pattern is spot pattern from single crystal and ring pattern from polycrystals. I don´t know what is your sample about, but it could be thin foil with oxidic layers on bottom and top surface. These tiny oxidic crystals are usually residuals from not well performed electropolishing and diffraction pattern suffer by signal from these oxides.
Another explanation could be that every TEM sample has got the amorphous layers on the surfaces and (in dependence of its composition) after irradiation with electron beam the amorphous layer goes easily to crystalline state.
Also it is possible to consider about more than one phase contributing to ring pattern. This claiming has low probability by me, because I recognize only few rings. In case of more phases there are much more rings.
It is very difficult to say what is the reason of your diffraction pattern appearance without additional information about sample preparation, sample type and form, observation conditions. I see huge amount of inelastic scattering in diffraction pattern so my guess is that your sample was very thick and diffraction rings are caused by oxide nanocrystals on the surfaces.
By the way, try to change camera acquisition settings and process the diffraction pattern in special software, e.g. Fit2D, which may help you to enhance the reflection intensities.