Another alternative is KMP (Knuth–Morris–Pratt).
The KMP algorithm searches for a length-m substring in a length-n string in worst-case O(n+m) time, compared to a worst-case of O(n·m) for the naive algorithm, so using KMP may be reasonable if you care about worst-case time complexity.
Here's a JavaScript implementation by Project Nayuki, taken from https://www.nayuki.io/res/knuth-morris-pratt-string-matching/kmp-string-matcher.js:
// Searches for the given pattern string in the given text string using the Knuth-Morris-Pratt string matching algorithm.
// If the pattern is found, this returns the index of the start of the earliest match in 'text'. Otherwise -1 is returned.
function kmpSearch(pattern, text) {_x000D_
if (pattern.length == 0)_x000D_
return 0; // Immediate match_x000D_
_x000D_
// Compute longest suffix-prefix table_x000D_
var lsp = [0]; // Base case_x000D_
for (var i = 1; i < pattern.length; i++) {_x000D_
var j = lsp[i - 1]; // Start by assuming we're extending the previous LSP_x000D_
while (j > 0 && pattern.charAt(i) != pattern.charAt(j))_x000D_
j = lsp[j - 1];_x000D_
if (pattern.charAt(i) == pattern.charAt(j))_x000D_
j++;_x000D_
lsp.push(j);_x000D_
}_x000D_
_x000D_
// Walk through text string_x000D_
var j = 0; // Number of chars matched in pattern_x000D_
for (var i = 0; i < text.length; i++) {_x000D_
while (j > 0 && text.charAt(i) != pattern.charAt(j))_x000D_
j = lsp[j - 1]; // Fall back in the pattern_x000D_
if (text.charAt(i) == pattern.charAt(j)) {_x000D_
j++; // Next char matched, increment position_x000D_
if (j == pattern.length)_x000D_
return i - (j - 1);_x000D_
}_x000D_
}_x000D_
return -1; // Not found_x000D_
}_x000D_
_x000D_
console.log(kmpSearch('ays', 'haystack') != -1) // true_x000D_
console.log(kmpSearch('asdf', 'haystack') != -1) // false