I've done a quick review of textbooks to clarify the issue. I hope this could be of assisstance:
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Boron WF, Boulbpaep EL, Medical Physiology, Updated 2nd Ed., Chapter 58, p. 1215: "...Most of the body’s fat depots exist in the subcutaneous adipose tissue layers, although fat also exists to a small extent in muscle and in visceral (deeper) depots in obese individuals..."
Op. cit., p. 1225: "Insulin, secreted in response to the carbohydrate or protein components of the meal, has three major effects on lipid metabolism (see Chapter 51). First, insulin stimulates LPL. LPL promotes hydrolysis of TAGs to FAs and glycerol. The breakdown products enter the adipocytes for re-esterification into TAGs. Insulin promotes storage in muscle and adipose tissue of both exogenous TAGs (derived from a meal and carried in chylomicrons) and endogenous TAGs (produced by the liver and carried in VLDLs)."
Op. cit., p. 1231: "...In contrast, oxidative metabolism of FAs is the major mechanism for supporting exercising muscle during prolonged activity, such as running a marathon, which is mostly the work of the slow-twitch type 1 muscle fibers (see Chapter 60)."
Op. cit., p. 1232: "Integrative Metabolism of Fasting [...] In the fed state and early in the fasting state, the brain derives essentially all its energy from oxidation of glucose because ketone bodies are not present and the blood-brain barrier is mostly impermeable to FAs. Most other major organs of the body (liver, skeletal muscle, heart) fill their energy needs at this time by oxidizing FAs... After an overnight fast, the decline in circulating insulin leads to a marked decrease in glucose uptake by insulin-sensitive tissues (e.g., muscle) and a shift toward the use by these tissues of FAs mobilized from fat stores."
Op. cit., p. 1234: "Lipolysis - Finally, the fall in plasma [insulin] after an overnight fast permits the release of FAs and glycerol from fat stores. This response appears to be more pronounced in visceral than peripheral fat depots. The decline in [insulin] and the ensuing lipolysis are sufficient to supply FAs to extracerebral tissues (e.g., muscle, heart, liver) for fuel and glycerol to the liver for gluconeogenesis. However, these changes are not sufficient to stimulate the hepatic conversion of FA to ketone bodies."
Guyton & Hall Textbook of Medical Physiology, 11th Ed., Chapter 68, p. 846: "Probably the most dramatic increase that occurs in fat utilization is that observed during heavy exercise. This results almost entirely from release of epinephrine and norepinephrine by the adrenal medullae during exercise, as a result of sympathetic stimulation. These two hormones directly activate hormone-sensitive triglyceride lipase, which is present in abundance in the fat cells, and this causes rapid breakdown of triglycerides and mobilization of fatty acids.Sometimes the free fatty acid concentration in the blood of an exercising person rises as much as eightfold, and the use of these fatty acids by the muscles for energy is correspondingly increased. Other types of stress that activate the sympathetic nervous system can also increase fatty acid mobilization and utilization in a similar manner."