Finding a scent within your budget that fits your criteria is going to be a tough call, unless you are willing to buy small decants, but do sample first.
The cheapest jasmine l can think of is Lust by Lush, but l find it very strong. Perhaps the solid stick version might suit you better.
For a "garden after the rain" scent, l highly recommend Un Matin d'Orage by Annick Goutal.
For a lighter jasmine, try Estee Lauder Private Collection Jasmin White Moss, By Kilian Love & Tears, or Guerlain Acqua Allegoria Jasminora.
Welcome to basenotes, & do report back on what you find!