diff --git a/ggml/src/ggml-cuda/ggml-cuda.cu b/ggml/src/ggml-cuda/ggml-cuda.cu
index 255fcf8294b5..2c2889c5f533 100644
--- a/ggml/src/ggml-cuda/ggml-cuda.cu
+++ b/ggml/src/ggml-cuda/ggml-cuda.cu
@@ -424,6 +424,8 @@ const ggml_cuda_device_info & ggml_cuda_info() {
 // buffer pool for cuda (legacy)
 struct ggml_cuda_pool_leg : public ggml_cuda_pool {
     static const int MAX_BUFFERS = 256;
+    // far below any KV-sized scratch, so small buffers are never rounded
+    static const size_t POOL_POW2_MIN_SIZE = 4ull*1024*1024;
 
     int device;
     struct ggml_cuda_buffer {
@@ -463,6 +465,7 @@ struct ggml_cuda_pool_leg : public ggml_cuda_pool {
 #endif
         size_t best_diff = 1ull << 36;
         int ibest = -1;
+        size_t largest_unusable = 0;
         for (int i = 0; i < MAX_BUFFERS; ++i) {
             ggml_cuda_buffer& b = buffer_pool[i];
             if (b.ptr != nullptr) {
@@ -470,7 +473,9 @@ struct ggml_cuda_pool_leg : public ggml_cuda_pool {
                 ++nnz;
                 if (b.size > max_size) max_size = b.size;
 #endif
-                if (b.size >= size) {
+                if (b.size < size) {
+                    if (b.size > largest_unusable) largest_unusable = b.size;
+                } else {
                     size_t diff = b.size - size;
                     if (diff < best_diff) {
                         best_diff = diff;
@@ -497,6 +502,20 @@ struct ggml_cuda_pool_leg : public ggml_cuda_pool {
         void * ptr;
         size_t look_ahead_size = (size_t) (1.05 * size);
         look_ahead_size = 256 * ((look_ahead_size + 255)/256);
+        const size_t exact_size = look_ahead_size;
+        // Nothing cached can serve this request, and a cached block smaller than it
+        // never will, so a caller that grows its request step by step strands one block
+        // per step. A largest cached block just below the request is that signature:
+        // round up to a power of two so the series reuses one block per octave instead.
+        if (size > POOL_POW2_MIN_SIZE && largest_unusable >= size - size/4) {
+            size_t pow2 = POOL_POW2_MIN_SIZE;
+            while (pow2 < size && pow2 <= SIZE_MAX/2) {
+                pow2 *= 2;
+            }
+            if (pow2 >= size) {
+                look_ahead_size = pow2;
+            }
+        }
         ggml_cuda_set_device(device);
         cudaError_t err = ggml_cuda_device_malloc(&ptr, look_ahead_size, device);
         if (err == cudaErrorMemoryAllocation) {
@@ -506,6 +525,7 @@ struct ggml_cuda_pool_leg : public ggml_cuda_pool {
                            device, look_ahead_size/1024.0/1024.0, cached_bytes/1024.0/1024.0);
             CUDA_CHECK(cudaDeviceSynchronize());
             clear_pool();
+            look_ahead_size = exact_size;   // retry without the rounding
             err = ggml_cuda_device_malloc(&ptr, look_ahead_size, device);
             if (err == cudaSuccess) {
                 GGML_LOG_DEBUG(GGML_CUDA_NAME " pool[%d]: retry succeeded\n", device);