Do we have "blades of hay" like "blades of grass"?

Solution 1:

One possibility is a stalk of hay, or haystalk:

stalk noun [ C ]
​
the main stem of a plant, or the narrow stem that joins leaves, flowers, or fruit to the main stem of a plant:

She trimmed the stalks of the tulips before putting them in a vase.

Cambridge Dictionary

Haystalk

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Related to Haystalk: hayrick, haystack, making hay

n. A stalk of hay.

The Free Dictionary

Solution 2:

I think it is called a hay straw:

• a single dry or ripened stalk, esp of a grass.

(Dictionary.com)

From AmancheThe Story of Japanese Internment in Colorado During World War II

• A Mercedian was shown in a "hick" costume, complete with a ragged straw hat on his head and a hay straw in his mouth, and remarking,
• The minute she was out, she went to find Sweet Daddy. He leaned nonchalantly on a bale of hay, picking his teeth with a hay straw. He refused to meet her eyes.