diff --git a/CHANGELOG.md b/CHANGELOG.md
index d017a90a15b..d2b8d81e01d 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -60,6 +60,7 @@
     - `zerotrie`
         - Add `as_borrowed_slice` and `AsRef` impl (https://github.com/unicode-org/icu4x/pull/4381)
         - Add `ZeroTrieSimpleAsciiCursor` for manual iteration (https://github.com/unicode-org/icu4x/pull/4383)
+        - Increase bound of `p` to solve more perfect hash functions; _might_ break serialized ZeroTriePerfectHash from previous versions (https://github.com/unicode-org/icu4x/pull/4888)
     - `zerovec`
         - Change `ZeroHashMap` to use `twox-hash` (https://github.com/unicode-org/icu4x/pull/4592)
     - `writeable`
diff --git a/utils/zerotrie/src/byte_phf/builder.rs b/utils/zerotrie/src/byte_phf/builder.rs
index 421bd7230bd..20a8eecba0e 100644
--- a/utils/zerotrie/src/byte_phf/builder.rs
+++ b/utils/zerotrie/src/byte_phf/builder.rs
@@ -26,7 +26,7 @@ pub fn find(bytes: &[u8]) -> Result<(u8, Vec<u8>), Error> {
 
     let mut bqs = vec![0u8; N];
     let mut seen = vec![false; N];
-    let mut max_allowable_p = P_FAST_MAX;
+    let max_allowable_p = P_FAST_MAX;
     let mut max_allowable_q = Q_FAST_MAX;
 
     'p_loop: loop {
@@ -67,13 +67,15 @@ pub fn find(bytes: &[u8]) -> Result<(u8, Vec<u8>), Error> {
                         num_max_q += 1;
                         bqs[i] = 0;
                         if i == 0 || num_max_q > MAX_L2_SEARCH_MISSES {
-                            if p == max_allowable_p && max_allowable_p != P_REAL_MAX {
+                            if p == max_allowable_p && max_allowable_q != Q_REAL_MAX {
                                 // println!("Could not solve fast function: trying again: {bytes:?}");
-                                max_allowable_p = P_REAL_MAX;
                                 max_allowable_q = Q_REAL_MAX;
                                 p = 0;
                                 continue 'p_loop;
-                            } else if p == P_REAL_MAX {
+                            } else if p == max_allowable_p {
+                                // If a fallback algorithm for `p` is added, relax this assertion
+                                // and re-run the loop with a higher `max_allowable_p`.
+                                debug_assert_eq!(max_allowable_p, P_REAL_MAX);
                                 // println!("Could not solve PHF function");
                                 return Err(Error::CouldNotSolvePerfectHash);
                             } else {
diff --git a/utils/zerotrie/src/byte_phf/mod.rs b/utils/zerotrie/src/byte_phf/mod.rs
index bcf77607968..b0cb6d09a0d 100644
--- a/utils/zerotrie/src/byte_phf/mod.rs
+++ b/utils/zerotrie/src/byte_phf/mod.rs
@@ -59,14 +59,16 @@ mod cached_owned;
 pub use cached_owned::PerfectByteHashMapCacheOwned;
 
 /// The cutoff for the fast version of [`f1`].
-const P_FAST_MAX: u8 = 11;
+#[cfg(feature = "alloc")] // used in the builder code
+const P_FAST_MAX: u8 = 95;
 
 /// The cutoff for the fast version of [`f2`].
 const Q_FAST_MAX: u8 = 95;
 
 /// The maximum allowable value of `p`. This could be raised if found to be necessary.
+/// Values exceeding P_FAST_MAX could use a different `p` algorithm by modifying [`f1`].
 #[cfg(feature = "alloc")] // used in the builder code
-const P_REAL_MAX: u8 = 15;
+const P_REAL_MAX: u8 = P_FAST_MAX;
 
 /// The maximum allowable value of `q`. This could be raised if found to be necessary.
 #[cfg(feature = "alloc")] // used in the builder code
@@ -118,13 +120,10 @@ pub fn f1(byte: u8, p: u8, n: usize) -> usize {
     if p == 0 {
         byte as usize % n
     } else {
-        let mut result = byte ^ p ^ byte.wrapping_shr(p as u32);
-        // In almost all cases, the PHF works with the above constant-time operation.
-        // However, to crack a few difficult cases, we fall back to the linear-time
-        // operation shown below.
-        for _ in P_FAST_MAX..p {
-            result = result ^ (result << 1) ^ (result >> 1);
-        }
+        // `p` always uses the below constant-time operation. If needed, we
+        // could add some other operation here with `p > P_FAST_MAX` to solve
+        // difficult cases if the need arises.
+        let result = byte ^ p ^ byte.wrapping_shr(p as u32);
         result as usize % n
     }
 }
@@ -359,6 +358,24 @@ mod tests {
         assert!(count_fastq >= count_slowq * 100);
     }
 
+    #[test]
+    fn test_hard_cases() {
+        let keys = [
+            0u8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
+            24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
+            46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67,
+            68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
+            90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108,
+            109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125,
+            126, 195, 196,
+        ];
+
+        let computed = PerfectByteHashMap::try_new(&keys).unwrap();
+        let (p, qmax) = computed.p_qmax().unwrap();
+        assert_eq!(p, 69);
+        assert_eq!(qmax, 67);
+    }
+
     #[test]
     fn test_build_read_small() {
         #[derive(Debug)]