By far the most essential yet difficult resource to get when starting in Summer particularly is grass, second to which is nitre. If you start in Summer and stay on surface, both of these resources you can really only get from the meteor areas and tumbleweeds in desert. One idea is to add another grass resource variant, the Grassy Trees in Savanna biomes which would shed grass as they grow or could be chopped for grass and replanted for more.
Really feels like this season as a whole should be less challenging. Winter is a cool challenge and gives you new stuff to work with, but Summer is just hated in general besides living through it in caves, even with the good stuff you can get only during this season on surface, and that's in part because in winter you can just burn a tree to heat up and stay warm for up to a minute with nothing on you, while in Summer you don't have an option as easy as that. May be you would not overheat or have plants wither normally unless there is a heatwave happening, which could be intermittent like rain in Spring and items like the Ice Cube could be made better and more reliable if you gathered the ice (slower wetness gain, no speed reduction).
I think you also mixed up the starting items for Spring and Summer. You get straw hat for Spring, but pretty parasol for Summer. Both work for both seasons, but clearly Straw Hat is more valuable, fitting and needed for Summer than the Pretty Parasol, and the same for the parasol in Spring (bar the valuable part). You don't need to gather any grass to get one for the season and can use whirly fan as well, but the parasol just ruins that option unless you started near the end of Spring. The hat also takes 12 grass to craft, which is a hefty price when you can't even gather a single one in your vicinity.